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Операция «Дина́мо» (Дюнке́ркская операция, Дюнке́ркская эвакуация) () — операция в ходе Французской кампании Второй мировой войны по эвакуации морем английских, французских и бельгийских частей, блокированных у города Дюнкерк немецкими войсками после Дюнкеркской битвы.

Предыстория

После прорыва линии Мажино 10 мая 1940 года и капитуляции Нидерландов 14 мая немецкое командование развило свой успех. Части Британских экспедиционных сил под командованием лорда Джона Горта, французские части и соединения, входившие в 16-й корпус, и остатки бельгийских войск оказались блокированными в районе города Дюнкерк.

18 мая 1940 года командующий британскими войсками лорд Горт впервые открыто предложил рассмотреть вопрос эвакуации союзных войск на Британские острова.

После того как 20 мая 1940 года немецкие танковые соединения прорвались к Абвилю, войска 1-й группы армий союзников (в общей сложности, 10 английских, 18 французских и 12 бельгийских дивизий) оказались отрезаны и прижаты к морю в районе Гравлин, Аррас, Брюгге. С юга и юго-запада против них действовали войска немецкой группы армий «А» под командованием генерал-полковника Герда фон Рундштедта (танковая группа Э. Клейста, танковая группа Г. Гота и 4-я армия вермахта), с востока и юго-востока — войска немецкой группы армий «Б» под командованием генерал-полковника В. Лееба (части 18-й и 6-й армий).

Кабинет Черчилля и Британское Адмиралтейство решили эвакуировать части британского экспедиционного корпуса на Британские острова.

Ход операции
20 мая британское правительство начало собирать любые корабли и суда, которые были способны принять участие в эвакуации (см. Маленькие корабли Дюнкерка). Для эвакуации командование союзников мобилизовало все имевшиеся в наличии корабли военно-морского и торгового флотов: 693 английских и около 250 французских. Планировал и руководил операцией контр-адмирал Бертрам Рамсей.

21 мая 1940 года 19-й корпус вермахта получил приказ перейти в наступление с целью овладеть портами на Ла-Манше. В этот же день, во второй половине дня, британские войска предприняли контратаку против немецких частей в районе к югу от Арраса, в контратаке участвовали ограниченные силы (один пехотный полк и два танковых батальона). Подразделения 4-й армии вермахта при поддержке штурмовой авиации ликвидировали угрозу, однако немецкие войска были отброшены на несколько километров к югу.

В ночь на 22 мая 1940 в контратаку перешли две французские дивизии, но из-за отсутствия координации действий между союзниками к этому времени английское командование уже остановило продвижение и дало приказ своим войскам отойти. Контратака, получившая название «Кризис под Аррасом», вызвала переполох у немецкого командования. В 1945 году Рундштедт писал:
«Критический момент наступления возник как раз в то время, когда мои войска достигли Ла-Манша. Это был контрудар английских войск, нанесённый 21 мая к югу от Арраса. В течение короткого времени мы опасались, что наши танковые дивизии будут отрезаны, прежде чем успеют подойти на помощь пехотные дивизии. Ни одна из французских контратак не представляла такой серьёзной угрозы, как эта».

22 мая 1940 года части танковой группы Клейста заняли Булонь. В этот же день английское военно-морское министерство реквизировало 40 голландских шхун, находившихся в английских портах, с целью использовать их для эвакуации войск с континента.

Как отмечал в воспоминаниях командующий 19-м армейским корпусом генерал Г. Гудериан, в течение этого дня в боях под Девром, Саме и южнее Булони военнослужащие корпуса сталкивались не только с англо-французскими войсками, но и с подразделениями бельгийских и голландских войск.

23 мая 1940 года немецкие части вышли к порту Кале и оказались в 16 километрах от Дюнкерка.

В ночь с 23 на 24 мая 1940 года 9-я рота полка «Der Führer» 2-й танковой дивизии СС, выдвинувшаяся в район Байёля для охраны переправ, была атакована силами противника численностью до батальона пехоты при поддержке танков, которые прорвались через оборонительные позиции роты. Одновременно французские танки атаковали позиции 7-й роты полка. На помощь 9-й роте в бой под Сент-Илером были брошены резервы 9-й роты, а также один пулемётный взвод из состава 12-й роты и один противотанковый взвод из состава 14-й роты полка «Der Führer».

24 мая Гитлер отдал приказ немецким танковым дивизиям, наступавшим вдоль побережья Ла-Манша, остановить наступление на рубеже канала Аа и отвести назад части, продвинувшиеся на Азбрук. Дальнейшее продвижение разрешалось только частям, выполнявшим задачи по разведке и охранению. В результате немецкие части остановились на рубеже Бетюн — Сент-Омер — Гравлин. Гитлер приказал «не приближаться к Дюнкерку ближе чем на 10 км» и не использовать против блокированной группировки танки, поэтому Рундштедт, стараясь воспрепятствовать эвакуации, но не нарушить полученный приказ фюрера, отдал немецким войскам приказ использовать для обстрела позиций противника артиллерию среднего калибра.
В этот же день, в 11 часов 42 минуты, незашифрованное сообщение, в котором немецкое военное командование приказало войскам остановиться на линии Дюнкерк — Азбрук — Мервилль, было перехвачено английской службой радиоперехвата.

Тем не менее, 24 мая по приказу командира дивизии СС «Адольф Гитлер» военнослужащие дивизии переправились через канал Аа и взяли высоту Монваттан на противоположном берегу, которая обеспечивала господство над равнинной местностью (развалины средневекового замка на вершине позволяли превратить её в опорный пункт).

Вечером 26 мая британский экспедиционный корпус получил приказ об эвакуации. Непосредственно перед началом операции правительство Великобритании обратилось ко всем владельцам частных кораблей, катеров и иных судов с призывом принять участие в эвакуации войск.

27—28 мая полк «Der Führer» 2-й танковой дивизии СС вёл бои с подразделениями британских войск в районе леса Ньеп за переправу через реку Лис. Боевые действия носили ожесточённый характер и доходили до рукопашных схваток, лесной массив Ньеп был занят немцами только к вечеру 28 мая.

28 мая 1940 года бельгийский король Леопольд III подписал акт о капитуляции Бельгии. Капитуляция бельгийских войск высвободила немецкие воинские части и осложнила положение англо-французских войск, блокированных в районе Дюнкерка.

Эвакуация из района Дюнкерка проходила рассредоточенно, под непрерывным огнём артиллерии и постоянными бомбардировками противника. Особо массированные налёты начинались после отхода на заправку английских истребителей, прикрывавших побережье. При сужении кольца окружения огонь вёлся и из стрелкового оружия, в первую очередь из пулемётов. Погрузка войск на крупные корабли британского военно-морского и торгового флота проходила в порту Дюнкерка, однако войска на побережье создали нескольких импровизированных причалов из загнанных в воду колонн автомашин, к которым могли швартоваться небольшие корабли британского вспомогательного флота. Кроме того, под прикрытием кораблей британского военно-морского флота мелкие корабли и катера подходили к побережью, и солдаты добирались до них на лодках, спасательных шлюпках и самостоятельно изготовленных плавсредствах.

Боевые действия велись в условиях сильно пересечённой местности среди многочисленных каналов, английские войска удерживали восточную часть фронта, французские — западную; по мере эвакуации войск передовые части выходили из соприкосновения с противником и отправлялись на берег для погрузки, на следующем рубеже обороны противника встречали новые части арьергарда. Немцы постоянно атаковали, но несли тяжёлые потери и продвигались чрезвычайно медленно. Иногда союзные войска контратаковали и отбрасывали их на исходные позиции.

Самолёты немецких военно-воздушных сил не сумели достичь превосходства в воздухе и сорвать эвакуацию войск союзников по ряду причин, среди которых:
 переоценка командованием люфтваффе сил и возможностей немецкой авиации (генерал А. Кессельринг отмечал, что при постановке задачи Геринг не учитывал степень усталости лётчиков люфтваффе, которые в течение почти трёх недель участвовали в боевых действиях), а также недостаточный учёт местных особенностей театра военных действий:
 так, при бомбардировке побережья поражающее действие немецких авиабомб оказалось снижено в результате низкой плотности морского песка;
 активное противодействие со стороны истребительной авиации противника в период проведения операции (только самолёты английских военно-воздушных сил выполнили 2739 самолёто-вылетов над зоной эвакуации);
 в частности, активные действия английских истребителей Supermarine Spitfire, затруднившие люфтваффе выполнение поставленных перед ними задач
 сохранявшаяся в течение нескольких дней плохая погода, которая не препятствовала эвакуации, но затрудняла действия авиации.

Результаты
По официальным данным военно-морского министерства Великобритании, в ходе операции «Динамо» (в период с 26 мая по 4 июня 1940 года) с французского побережья в районе Дюнкерка было эвакуировано в общей сложности 338 226 военнослужащих союзников. Из этого количества до начала операции «Динамо» из района Дюнкерка на Британские острова было эвакуировано 59,3 тысяч британских военнослужащих, в ходе операции «Динамо» были вывезены ещё 139,8 тыс. англичан и 139 тысяч военнослужащих стран-союзников (около 90 тысяч французов, а также бельгийцы и военнослужащие других союзных стран). Джон Даунинг был награждён  орденом «За выдающиеся заслуги».

Некоторое количество военнослужащих погибло в ходе транспортировки.

На Британские острова из Франции прибыло 337 131 человек.
Дюнкеркская операция позволила сохранить кадровую британскую армию, получившую при этом бесценный боевой опыт, хотя армия и лишилась практически всего тяжёлого вооружения. Был сохранён весь личный состав, ставший в дальнейшем основой войск союзников. Английское командование перед началом эвакуации рассчитывало, что сможет спасти всего около 45 тысяч человек, но в ходе упорных боёв британские вооружённые силы продемонстрировали высокий боевой дух и профессионализм. Около половины войск были спасены гражданским населением, рыбаками, паромщиками, владельцами яхт, катеров и другими, откликнувшимися на призыв британских властей. Наравне с англичанами также действовали многие французы, бельгийцы и другие союзники, те, кто не поддался панике и не был заражён пораженческими настроениями. Многие из них в дальнейшем продолжили сражаться в составе как британских вооружённых сил, так и в военных формированиях, таких как «Свободная Франция», решивших, несмотря на капитуляцию своих правительств, продолжать борьбу.

Потери

Антигитлеровская коалиция
При эвакуации личного состава английских, французских и бельгийских войск в районе Дюнкерка было брошено практически всё тяжёлое вооружение, техника и снаряжение. В общей сложности, было оставлено 2472 артиллерийских орудия, почти 65 тысяч автомашин, 20 тысяч мотоциклов, 68 тысяч тонн боеприпасов, 147 тысяч тонн топлива и 377 тысяч тонн снаряжения и военного имущества, 8 тысяч пулемётов и около 90 тысяч винтовок, в том числе всё тяжёлое вооружение и транспорт 9 английских дивизий. Потери Королевских военно-воздушных сил, прикрывавших эвакуацию, составили 106 самолётов. В ходе операции и при транспортировке в Англию погибло и пропало без вести около 2 тысяч солдат и моряков союзников.

В ходе боёв с немецкими войсками во время проведения операции «Динамо» и после её окончания в районе Дюнкерка были пленены в общей сложности 50 тысяч военнослужащих французской армии. Несмотря на то, что английский военно-морской флот до последней возможности не прекращал эвакуацию и успел вывезти более 26 тысяч французов, около 15 тысяч военнослужащих французской армии попали в плен к немцам, когда прикрывали последний этап эвакуации, будучи в арьергарде эвакуирующихся войск. Впоследствии немецкая пропаганда использовала этот эпизод для разжигания антибританских настроений среди французов.

В ходе операции были потеряны более четверти участвовавших в эвакуации кораблей и судов (224 английских и около 60 французских судов), в том числе 6 эсминцев английского военно-морского флота и 3 французских ВМС. Значительное количество кораблей получили повреждения (в том числе, 19 или 23 корабля английского военно-морского флота).

Третий рейх
Германские войска потеряли 140 самолётов в воздушных боях и от огня зенитной артиллерии.

Людские потери составили 8,2 тысячи человек.

Причины остановки наступления немцев
Об истинных причинах остановки немецкого наступления по приказу Гитлера военные историки спорят и по сей день. Были выдвинуты предположения, объясняющие это решение:

 Гитлер стремился избежать войны на два фронта. С идеей «Жизненного пространства на Востоке» война с СССР была неизбежна. Летом 1940 года Гитлер прямо говорил, что нет причин для дальнейшего продолжения войны между Германией и Великобританией (предложение Гитлера от 19 июля заключить мир с Германией), и призывал сесть за стол переговоров тем самым пытаясь заключить мир или же временное перемирие для переброски основных сил на готовившийся Восточный фронт (План «Барбаросса»)
 Гитлер стремился избежать дополнительных потерь в танковых частях, которые должны были вновь вступить в бои против французских войск во второй фазе французской кампании. При этом командующий 19-м армейским корпусом генерал Г. Гудериан (непосредственно участвовавший в операции по окружению англо-французских войск в районе Дюнкерка) считал, что «нервозность» Гитлера по этому вопросу являлась необоснованной, и что немецкие войска могли бы уничтожить блокированную группировку.
 начальник генерального штаба сухопутных войск Ф. Гальдер не рекомендовал атаковать плацдарм, предупреждая об опасности возможной контратаки французских войск из района Парижа.
 переоценка возможностей ВВС — шеф люфтваффе Геринг обещал фюреру, что сможет легко предотвратить эвакуацию с помощью ВВС, а танкам необходимо повернуть на юг и завершить кампанию против Франции.
 Гитлер надеялся заключить с Британией мир на выгодных условиях и намеренно помешал ликвидации войск, что должно было облегчить ему эту задачу.
 Гитлер или представители командования вермахта опасались наступления французских войск на других участках фронта и предпочли уменьшить потери от наступления на позиции противника.

Объективно стоит отметить — после войны многие немецкие генералы стремились переложить ответственность за свои неудачи на Гитлера. Тем не менее, согласно имеющейся информации, можно сделать вывод: генеральный штаб и большинство участвовавших в кампании военачальников опасались контрнаступления французской армии из района Парижа и встречного удара с побережья британских экспедиционных сил.

Ещё одна причина остановки наступления: потери в танках и пехоте, достигавшие к этому моменту от 30 до 50 %. Отсюда опасения немецкого командования полностью потерять танковые войска при попытке атаковать отрезанных союзников — в таком случае продолжение войны на континенте против французской армии, тогда ещё далеко не обескровленной, стало бы невозможным. Только после того, как части вермахта были усилены подтянутой пехотой и артиллерией, и стало окончательно ясно, что французская армия дезорганизована, наступление немецких войск было возобновлено.

В любом случае, приказ о ликвидации окружённой группировки был отдан поздно, союзные войска успели укрепить свои позиции и провести эвакуацию, силы люфтваффе не справились с задачей им помешать, а англичане воспрянули духом благодаря «чуду Дюнкерка» и только укрепились в решимости продолжать войну.

Отражение в культуре и искусстве
 Главный герой повести «Белая гусыня» (1940) американского писателя Пола Гэллико отправляется через Ла-Манш на лодке к Дюнкерку, чтобы спасать своих соотечественников, и гибнет, перевезя сотни солдат на ожидающие их суда.
 Эвакуация описана в романе Робера Мерля «Уикэнд на берегу океана» (1949) и показана в его одноимённой экранизации (1964) Анри Вернёем с Жан-Полем Бельмондо в главной роли.
 В 1958 году в Великобритании вышел фильм «» режиссёра Лесли Нормана, сценарий которого был основан на одноимённой книге подполковника Юэна Батлера и майора Дж. С. Брэдфорда, а также на романе «The Big Pick-Up» (1955) Эллестона Тревора.
 «Штуки» над Лондоном" 1969, совместное производство Испания, Италия, Франция
 Эвакуация описана в романе Иэна Макьюэна «Искупление» (2001) и показана в его одноимённой экранизации (2007) Джо Райтом.
 Студия Warner Bros. Pictures 20 июля 2017 года выпустила фильм «Дюнкерк», описывающий события эвакуации; режиссёр Кристофер Нолан.
 Затронута в британском драматическом байопике режиссёра Джо Райта «Тёмные времена» (2017).

Примечания

Литература

На русском языке
 Гальдер Ф. Военный дневник: Ежедневные записи начальника Генерального штаба Сухопутных войск 1939—1942 гг. — Т 1: От начала войны с Польшей до конца наступления на Западном фронте (14.8.1939 г. — 30.6.1940 г.) — : Воениздат, 1968.
 Дивайн Д. Девять дней Дюнкерка / Сокр. пер. с англ. — : Воениздат, 1965. — 239 с.
 Сквайрс Р. Дорога войны. — : Издательство «Литературной газеты», 1952.
 Комплеев А. В. Дискуссия о причинах «дюнкеркского чуда» в отечественной историографии 1950-х — 1970-х гг.  // Известия высших учебных заведений. — Поволжский регион. — Гуманитарные науки. — 2015. — № 4(36). — С. 14—24.

На других языках
 Groehler O. Menetekel Dünkirchen // «», № 6, 1961.

Ссылки

 Spitfires Join the Fighting - aerial battle over Dunkirk
 Official website of Dunkirk memorial and museum
 BBC Archive - Dunkirk Evacuation
 BBC Archives — J. B. Priestley's 'Postscript' - radio broadcast from 5 June 1940

История Дюнкерка
Французская кампания
Эвакуации
Конфликты 1940 года
Май 1940 года
Июнь 1940 года


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text:
Q&A: Catching up with Saint Louis football coach Cal Lee
By Paul Honda on January 23, 2019 | Featured, Football, Saint Louis football
Saint Louis coach Cal Lee has the Crusaders on a 26-game winning streak. Photo by Jamm Aquino/Star-Advertiser.
The announcement about St. Francis closing its high school (and middle school) doors at the end of the academic year has a far-reaching ripple effect.
Several private schools informed the students of St. Francis about application fees that will be waived, and more. Cal Lee, coach of three-time defending Open Division state football champion Saint Louis, would seem to be a possible option for St. Francis standouts Faaope Laloulu, Bubbah Aina-Chaves and Shepherd Kekahuna, among others including public schools. Tuition cost is a huge factor.
Laloulu, a 6-foot-7, 340-pound left tackle, has scholarship offers from Arizona, BYU, Hawaii, Oregon, San Diego State and USC. Aina-Chaves, an explosive RB/LB, has rushed for 2,935 yards and 32 TDs in two seasons. He has averaged 9.4 yards per carry.
Lee chatted with Hawaii Prep World on Wednesday afternoon.
HPW: After talking with Ope and Bubbah, it sounds like nothing is really set for them after this school year.
Lee: I'm sure they've got to finish up the school year and it's hard to talk about where you will be, but it's something you have to plan.
HPW: Bubbah has mentioned that he prefers to be in an environment with no distractions away from school.
Lee: Once you get off the campus, that's so important.
HPW: There have been other student-athletes who came to Saint Louis after their schools closed doors.
Lee: It's too bad that all these players have to break up and go their separate ways after playing together, but sometimes things happen for a reason and good things will happen no matter where they go. These players will help any school they decide to go to after coming of an undefeated season.
HPW: Can you talk about what the process would be like for someone applying at Saint Louis, transferring in from St. Francis? There are a lot of schools that are providing some help at least in the application process.
Lee: The waiving of the application fees is something, but somebody said they were going to get the same cost at another school that they had at St. Francis. If it's $10,000 and the school was $15,000, that doesn't make sense.
HPW: The costs of private-school education keep going up. In the old days, back a half-century, some private schools waived tuition for three-sport athletes who worked on campus. Now, nobody gets a break like that.
Lee: I don't know anything about it. Whatever the tuition is, you'll pay, and there's financial aid and need, that's for everyone. You've got to be able to come up with some money, it's not free. You don't want to get a kid and his family in a bind. If they can make it happen, fine. I can't tell a kid to come here and play football, my goodness. It's more than football. I think 'Iolani, Punahou, Saint Louis, if you want to come there, you've got to pay for it.
HPW: I don't know how parents do it except by getting a second and third job. Even with a 50-percent discount, that's still $800, $900 a month for tuition. Or more at the bigger schools.
Lee: Really honestly, I never recruited one kid to come here and get stuck financially. You try and help them like everyone else. If they can, they can. If they cannot, nothing wrong with that. I'd like to go to Stanford, I'd like to, but I can't afford it.
HPW: I"m hearing that tuition, room and board at USC is now $75,000 per year.
Lee: I don't know how they do it. I was fortunate, my boy (Jonathan) worked at his school (Oregon) and that helped. He didn't play, but he got an education and now he's teaching. Football doesn't last forever.
HPW: During the past few weeks and months, I've gotten to talk with a lot of football players about games and getting recruited. I've mentioned to some of them how you toss out big jugs of Muscle Milk to your guys.
Lee: Chocolate is big. They love chocolate.
HPW: There are also a bunch of Crusader football players who also play on the Division I basketball team for Coach Sol Batoon. Have you watched them play?
Lee: Oh yeah. Ben Scott, (Nick) Herbig, Liloa (Kapiko). He just signed with San Diego State. They were just up here. He was kind of waiting around. I told him, don't wait too long. The other guys playing basketball are Nalu (Liftee), Junior Wily and Jayden (de Laura). And AJ (Bianco). He's a freshman.
HPW: Who's the best basketball player among them?
Lee: I like Junior, he's played a lot of basketball. He played much more than the other guys.
HPW: What does your brother Ron (Lee, offensive coordinator) do in the winter?
Lee: He comes up and looks at film every day. From about 1 o'clock to about 3:30 p.m. He's here.
HPW: Even now? Would you say he is obsessed or addicted to football?
Lee: He's addicted and obsessed. I've known since we were kids. When I was like 9, he was always the guy watching. He's older than me, then we played against each other. He was at Saint Louis, I was at Kalani. I followed him up to Willamette. We've been there, came back, coached together at Kaiser, Saint Louis, UH and back to Saint Louis.
HPW: Plus the AFL Hawaiian Islanders and Kalani High School. So Coach Ron is there two, three days a week?
Lee: He is here five days a week.
HPW: Is he looking at a variety of video from all kinds of teams, or is it just Saint Louis?
Lee: He's looking at a variety of things, but it's all Saint Louis footage.
HPW: Does he watch individual receivers or backs on tape expecting improvement? Does he look for them to apply what they're being taught at practice? Or is he just looking at schemes and other things.
Lee: That's what he's doing. He talks to the kids and tells them what they've got to do to get better. He knows what they can do, but do they want to be the same or get better?
HPW: Especially when there are plenty of teams who might outwork the defending state champions this offseason.
Lee: We know what we've got to do.
HPW: So what happens if Coach Ron sees certain players not really taking instructions, not getting better at running a certain route, or executing a certain way?
Lee: They won't be playing, but it's fortunate that the kids really want to get better. They don't want to come here and kill time. You can see it in their faces. The bar is real high and the competition is there.
HPW: Do you see some of your guys on the field working out?
Lee: It's early and we've got some time off, and there's no organized practice, but we have a few kids that are on the field on their own throwing and catching. I'm not going to tell them not to do it. Some young ones and some older ones, not playing any winter sports. They're in the weight room.
notes January 23, 2019 7:58 pm
"Lee: Really honestly, I never recruited one kid to come here and get stuck financially."
look how specific this "coach" is. He flat out admits to recruiting his players. he may not have recruited one kid to go to STL and get stuck financially, but what about all the other kids he recruited with "financial aid" packages??? Of course STL was buzzing around st francis campus like vultures trying to pick off the best players and recruiting them. those kids should go back to their community schools where they belong!
this interview is flat out evidence of "coach" cal lee admitting to recruiting
??? January 23, 2019 8:29 pm
The only ones that don't believe St Louis Academy Recruits is in Denial.
Ask all the former all state players from Mayor Wright Housing how they was able to afford STL tuition and none of them would say my parents paid for it.. NONE !!
Why doesn't HPW just ask Cal straight up about his recruiting & settle this debate once and for all?
Marauder January 23, 2019 8:33 pm
Cmon coach all us parents know. Alumni offers me to pay my son tuition to play for you. Sounded good but my son gets playing time, enjoys the game, and no waiting. He gets a better education where he is at.
T103 January 23, 2019 9:27 pm
@ Notes @ ???. U guys are seriously crazy. U two, well theres more of you, but you two (b/c u posted on this blog) are soooo hung up on this whole ordeal. Who the heck cares? Like no, seriously? who in the world cares? Just you two, and whoever else posts up the same kinda rhetoric on similar stories. I think u guys might be creating more health problems for yourselves being so stressed out about this, its such a huge deal for u guys. private schools recruit, who the heck gives a rip. I'm happy you stuck to ur guns and never let your own kids go to a private school to play for them which might give them better recognition to get ANOTHER scholarship to a college to get a degree and further their education. Way to stand ur ground, but alas, u can't tell anyone else how to parent their kids so u mine as well just keep your thoughts to yourself. If you guys have no concept of doing whatever it takes to give ur kids a better chance at a better future then man, I'm super glad u aren't my parents. if you feel that for your own flesh and blood kids that its better to go to a certain school, then so be it. If your gonna call st Louis on "recruiting" then u best be naming every other dang private school by name cuz they all do it, unless you just got so much hate and biased towards St. Louis. At least own your bias and hate, if there is no bias or hate, then call all the private school out by name. But truth is, do u know how many "doors" open b/c of name? How many more opportunities in the world students get b/c of the school they graduated or are associated with? Thats just life man, ain't nothing u can do bout it. Bitch and moan all u want, but it ain't gonna change and just unfortunate that you didn't have that opportunity. For all those students who get to go to a private school, recruited or not, I wish you the best, make the most of it, for those students who "stay in their community" I wish u the best too. Just hope that the younger generation can be be ok with other peoples success unlike "Notes" and "???".
Good Story January 23, 2019 10:09 pm
Daaannnggg…looks like someone hit a nerve.
Coach C January 23, 2019 10:49 pm
Lol. Tosses out Muscle Milk. I have lived in public housing. No way a family on public assistance can pay $1500. @month for private school.
ILH January 24, 2019 6:56 am
Reasons for Stl.sucess:
*Great coaching ( not all ,jus a few)
*Top level athletes(some recruited ,some go there on there own)
*Athletes getting held back a year (Huge game changer)
Robocop January 24, 2019 8:42 am
Good morning cry babies. You say no family on public assistance can afford $1500/month well it isn't $1500/month. Plus I live a few minutes away from KPT and I see all the 40k SUVs come in and out of there. They can make sacrifices if they want to. You guys (or 1 guy) live for these St Louis articles so you can be anonymous and talk trash while they smoke your team on the football field in real life. Who gives a crap if Cal recruits or doesn't. If your school recruited and won titles you would not be complaining about "staying home". Recruiting isn't illegal. But keep crying I'm sure it will help your community somehow. Losers
Coach C January 24, 2019 9:38 am
Lol, I live a couple of minutes away from MWH,KPT,K4H,KVH, Halawa ,etc. The stores around them places accept EBT for cigs and beer. It's a good place to farm huge kids. All HS programs do it. When will they come and offer scholarships for intellectual gifted children?
Hawaii Spokesman January 24, 2019 11:24 am
put it this way if saint louis recruits, who cares, thats y they are national ranked and thats why they dominate high school football in hawaii. Just face reality people, you guys are more out there to criticize a school that represents the state of hawaii in a big way. Look at the national attention they get. Why not be happy for them, representing hawaii which puts recruiting for college and help other kids at other schools also. Supposed to be backing up the hawaii people not going against them regardless what school they from. Saint Louis helped lots of schools rise on a national level here on hawaii and made football more competitive here. Shame on all you people who out here to criticize.
kimo January 24, 2019 11:43 am
4 peat
Marist ways that is what makes champion!
KALAEPOHAKU 4 LIFE!
GA Bulldog January 24, 2019 12:02 pm
Isn't Notes, ???, Go Figure and ILH the same person? Small kine schizophrenia but its all good whatever works for all of you!
Getting WHOOPED! January 24, 2019 3:05 pm
Recruit dis, Recruit dat! Cry all u like. All u adults that got whooped by St. Louis in the 80s/90s are now going to witness your son/sons friend/nephew/neighbors son etc get the same kind of whooping y'all took and theirs nothing y'all can do about it. Lol. Next 5 years at least the Koa Trophy stays at Kalaepohaku.
Recruiting is legal & fine in the ILH but it's the few Cocky alumni that come on here & brag about beating up on public schools or lesser ILH teams.
STL recruits & holds kids back a grade to get bigger, faster, stronger so you should win every year no surprise!
It's funny how you never hear those same cocky people bragging about beating Bishop Gorman, De La Selle & St John Bosco? Oh wait, that's because those teams wooooooooooped STL; why? Because they also recruit, and hold kids back legally!
Go figure, NUFF SAID !!!!
T1034life@yahoo.com January 24, 2019 6:23 pm
Dear ???, once again just own up to your strong dis-like of St. Louis, man up brah, and just say u biased and we can be done with it. If the article above already stated for ur satisfaction about recruiting, then please admit for yourself that u hate st Louis through and through. But also let it be known that I know students from all diff private schools who got held back for whatever reason. Ask around from those who attended, or attend or send their children to private school, it's NOT un-common for a kid. guy or girl, to be held back one year wen entering private school from public school. If u test into 9th and ur 8th grade ,they often hold um back a year, even in elem school, I've heard this scenario plenty of times so get a freakin grip man gee wiz. It's not only with sports, so what?!, the parents who hold their kids back in elem repeating a certain grade over in private school, are u gonna go ahead and say that they held them back so they could get smarter then their other classmates so they can win the math bowl for the 10th time in a row, or bring the science trophy back to the campus? seriously, whats wrong with you? HUGE CONSPIRACY, call a press conference, take an ad out on the radio and television and newspaper! Plead your cause! DO WHATEVER IT TAKES SO THAT ALL MAY KNOW THAT ST LOUIS AND ONLY ST LOUIS AND NO OTHER PRIVATE SCHOOL RECRUITS OR HOLDS ANY OTHER STUDENTS BACK IN ORDER FOR THEM TO GROW BIGGER, FASTER, STRONGER, AND KEEP THE TITLE ON THE SLOPES OF KALAEPOHAKU FOREVER!!!! AND THAT ALL OTHER SCHOOLS MAY GET BEATEN UP BY ST LOUIS AND ONLY ST LOUIS IN FOOTBALL BUT LETS NOT TALK ABOUT KAHUKU RECRUITING FOR BASKETBALL, BUT THATS NONE OF MY BUSINESS!!! LOL (how u figure, a student somehow enrolls at Kahuku for the time period of basketball season only, and then somehow "transfers" back to a different state or country for that matter?? But hey, none of my business yea?
suspense January 25, 2019 6:44 am
my daughter in grade 4 in a private school switched to another private school, which accepted her only as a grade 4 again. all because of her age; nothing to do with athletics or academics. their criterion was more appropriate to be within age group. at both schools, she was honor student.
Just Wondering January 25, 2019 12:06 pm
Just wondering…why would a kid, who lives in the Kahuku area, want to get up at 5:30 in the morning to travel into town to go to school, play sports, then end up driving back to the Kahuku area at 7 pm, stay up and study, and finally go to bed around 10, only to do it again in the morning, when he could just get up at maybe 7, and go to his high school in the area, that starts class at 7;45?? On top of that, have to fork out for gas and tuition, when he can go to a school that has no tuition…is it due to a choice that he and his parents make, or is he strapped to a tree and beaten to a pulp to be forced to go to that school in town (whether it be Punahou, Saint Louis, Kamemehameha)..just wondering…
@ just wondering, I think it's a conspiracy, I think they are being forced by the powers that be. It's against their will… we must help all those young kids who have no other choice but to attend private schools in town against their wishes, hopes, and dreams…
@ T1034life@yahoo.com
Geez, why you neva just write one book already😂😂😂😂
Sorry I hit a nerve but the truth hurts!
See you next Football season 🤙🏽
notes January 25, 2019 10:33 pm
LOL "kids get held back all the time!" so funny how at STL, seems like the only kids that get held back are on the football team. but must be a coincidence, right? it's not like we didn't have "coach" cal lee on this blog flat out admitting that he is destroying high school football and public school by recruiting for STL. "Coach" cal buys championships. FACT.
??? January 26, 2019 9:48 am
Why don't you tell Cal Lee to "Man up" and win a state title coaching at an OIA school ?
@Notes
You on point again!
Nobody from the STL Band has ever been held back to get better at playing the Flute or Tuba for an extra year.
😂😂😂😂😂😂😂😂😂
CynHyn January 26, 2019 6:25 pm
Lol, basketball Kahuku? Keep on track haters.
now that all the ILH haters have hard evidence of "coach" cal lee recruiting and destroying high school football, they all of a sudden got REALLY quiet. where you guys at? oh yeah. recruiting from our neighborhoods and backyards to keep your private "schools" open
T103 January 26, 2019 10:05 pm
@notes @???. You know what hits my nerves is the fact that your nerves get hit every single article that even mentions st. Louis period. I can bet my life savings that there will be some type of comment along the lines of 1. st Louis cheats.. 2. st Louis recruits. 3. st Louis buys titles or something along those lines. absolutely right "FACT" that it bothers me, ur like a broken record cuz u guys are so mento,. Only reason I chime in cuz you ALWAYS buys always post some comment about it. And u never even say out loud or consider that all the other private schools recruit as well, maybe not for football but for other sports as well. I swear. u can't even say that Punahou doesn't recruit. somehow ur flavor of hateorade is strictly St. Louis flavor. Ask any faculty member at ANY private Schoo if students of all ages get held back and then ask them if it's strictly to play sports, and then ask st Louis if every single dang student that gets held back is STRICTLY for football? The answer is freakin NO as a previous poster mentioned earlier. And I mention Kahuku basketball program b/c the kukai show the past couple of years, w the "fact" that students transfer from other countries right before basketball season and then magically disappear back to their country after basketball season is finish, coincidence? I would LOVE to hear you approach the parents in your community who send their kids private school, what would u say to them? and wen they reply that they just are doing what they think best for their child, what u going say bra? Take it a step further, call them out by first and last name on this blog? throw them under the bus! make your cause known u donkey.
1. private schools probably do recruit, what u going do about it? Nothing, all u going do is grumble.
2. Parents send their kids to whatever school they think is gonna give them a better opportunity for their future what you going do about it? nothing, cuz u can't do crap about it, that ain't ur freakin kid, so get over it.
Thats why it strikes a nerve with me, cuz you guy consistently say the same argument over and over again, article after article but u can't do kukai about it, and u can't even just generally say private schools as a whole recruit for whatever reason u only pin point St. Louis. Not once did u even mention Punahou and their dynasty of boys Vball titles, or punahou and their illustrious titles of track & filed, or swimming & diving. if you all gonna hate, at least hate fait and square and equally all around.
I will give you this, Kahuku has the most amazing fan base I have ever seen in my life, u guys will travel across the island to support ur teams. But the fact that you can't even be supportive for one of your own from your community doing something that they think will better themselves for their future & the future of their families, or even be happy for their success with another team thats not your own is just sad. Your love for them isn't unconditional, it's totally based on condition if they play for your community or not.
oh! T103! you must be back from your recruiting trip with "coach" cal! I hope you found more athletes to poach from not only community schools like Kahuku or mckinley or any other OIA school but st francis too. the fact is every school recruits but nobody did more to destroy the competitive balance of high school football than your best friend "coach" cal and he even admits it in this article. FACT
FACT: Punahou recruits because of course they do. they started recruiting in the mid 2000s because they know that the whole "education" excuse for private school is all a sham. private "schools" in the ILH can only stay open because of the money the football teams bring in. they gotta recruit or why the hell would anybody pay all that money just to go to some "school" with a sorry football team? there's more to school than just academics
FACT: the ILH should disband. send all the kids back home to their community schools and refund them all the tuition money their parents wasted on "education." let them know the truth that when you play for kahuku, you earn your championship rings for yourself, your family, and the community you grew up in. we rooted not recruited.
FACT: "schools" like STL talk about "Brotherhood" but all they do is buy players and championships.
FACT: STL not winning every year even though they steal the best talent in the whole state shows you how good the coaching staff is there
Punahou football recruiting goes way……………………………..back; probably, before you were born.
T103 January 27, 2019 10:01 am
@notes. I personally don't even care about sports tbh, I never played so it was never my priority. I don't even know him personally, never met him, never even said hi b/c I have never come in contact with him, if I saw him on the street I would recognize him but no communication with him whatsoever. Glad any student in any sport at any school gets a chance to learn discipline and work ethic, and team dynamics etc etc. Heres the deal, I'm sorry if you disagree, but thats a narrowed view of private schools. To say that all private schools are pushing their football programs with recruiting to win titles just for the sake of staying open is really ridiculous. Iolani, mid-pac, Kamehameha, sacred hearts, Damien, HBA, Le Jardin, all ain't pushing football programs. It IS about education, I'm sorry but if u drop the name of the private school that you graduated from, it opens doors and gets you connections, thats just the way it is in the real world after high school. Drop the name of Punahou or Iolani or Kamehameha and you get connections, try drop the name Kahuku and all your associated with is football. Thats not fair though bc majority of the student body at Kahuku isn't a football player, same thing with St. Louis, majority of the student population isn't on the football team, so much more to the students at Kahuku & St. Louis then just football. And sad to read your line about " theres more to Schoo then just academics". I think every good educator and parent knows its the opposite way around. Theres more to school then just sports. Of all the high school athletes, majority of them don't go on to play college, and of those that do, a fraction go on to play professionally, so what happens then? You have nothing to fall back on if you don't have a solid foundation of education, or just connections from name dropping per say. For you, is winning high school football titles all that it is? Can you confidently say that there isn't anything more then a balanced high school football system in hawaii? The kids that choose to go St. Louis, play on the team, get recognized by colleges, get scholarship offers from good EDUCATIONAL schools, might not have the same offers if they were to stay in their communities and play for their own local schools. But I'm sorry that you can't see that. As a parent, shouldn't you do anything and everything to give you child the best opportunity for his or her future?
By far this is your train of thought and your angle that you always come from in any of your comments and I'm really bummed that you can't see beyond this.
@T1034life@yahoo.com
Look at my posts from years ago, I have always said ALL ILH recruiting is unfair!! Not just STL.
But you prove you are the Narrow minded one by saying kids will get further in life with ILH hook ups? So if you come from Waianae or Kahuku you're screwed? C'Mon!!! Like no college will recognize you if you don't play for an ILH sports team? C'Mon. If you no like other peoples opinions don't join a blog🤷🏽♂
LOL your anti-public school bias is showing. "education." like the "education" the kids get when they go mercenary and learn that loyalty and community pride doesn't matter. family doesn't matter. splitting communities doesn't matter. because at the end of the day, all you do is name drop your private "school" that is doing all it can to destroy public school football and you will get a great job. but not like private school! idk about you but that's a sad sad world
@notes Lets me be fair, its up to the individual student to make it what it is at any school. Absolutely there are success stories from both private and public schools, there are non-success stories from both private and public schools. Students who go to private schools for majority understand that it costs a lot of $ for their parents to send them, and for the most part have better opportunities b/c of those connections, thats just the way it is, I don't even know what to tell u. I know students from both private & public with huge success and not so great lives after graduation. So what do u say to the student athletes who went private school, made it to professional level, and came back and gave back to their communities? Did that split up the community? Was that charity case? Is it a case of "oh poor them, let me just throw all of them people a bone"? It's not about name dropping, its about the fact that connections pay a huge role, and part of it is what school u graduated from. Like it or not, if a person is looking to hire someone, they will play more close attention to an applicant who graduated from the same school then a diff school, thats life plain and simple. And if you are saying that private schools are doing everything in their power to destroy public school football thats absurd. Thats actually laughable. its a combo of both doing sports at a school that has a name that is recognized to play at the next level and the connections u get from graduating there, sorry u feel the victim, it is the way it is. I'd be interested to know if u would completely pass if ur child was given the opportunity to go to a private school tuition free, and play whatever sport to possibly be recruited by a college that your kid would get to go for free for 4 years? Free education for 8 full years, would u pass it up ? or would u rip apart ur community by accepting that offer?
@??? If you go to a private school ur chances increase for connections thats all I'm saying. There are brilliant students are both private and public, but to be honest, top companies tend to look at private schools over public schools. Thats life…..
WOW, where do you get your information from? "Top companies look at private schools more than public"😂 You must have really went to STL to think like that😂
Companies look for College degrees, work experience or who you know! (Real World)
What rock have you been living under? You can have a G.E.D and still have a college degree so tell me how high school education is relevant in that situation??
T103 you keep bringing up irrelevant things. FACT: "Coach" Cal Lee just admitted to the entire planet that STL engages in recruiting. this recruiting is and always has been targeted by ILH to raid the public school communities of their best football players to raise the prestige of the ILH schools. who cares if their mercenary players go to college come back and do whatever.
you and your family has never gone through a recruiting battle and how the families get torn apart when the recruiters come with their scholarship offers and how the community is stung by losing to a school filled with your own players. you never gone through the experience of having to choose some b.s. promise of "education future" and proudly representing your hometown and the people and family you are closest to.
FACT: the ILH doesn't care about that. "coach" cal just recruit recruits recruits.then he laughs at us in interviews like this, then he uses brainwashed ILH people to defend the corrupt system he created. you have public school families cheering against the schools they played for. this is the football that ILH wants
network January 29, 2019 7:32 am
These Kahuku haters is hilarious. And whats funny is, these guys complaining probably never even played lol. So sad. Jealous Jealous jealous. Saint Louis will repeat to a 6 peat lol. Please keep smashing these OIA schools and make it a traditional thing again so the pain goes away for these guys and they get used to it that Saint Louis , yes dominates hawaii football , not kahuku. Kahuku trying to live in saint louis's shadows but cant , infact cant even come close. lol
Just wondering why I never hear people, who share the same view as you do, ever crucify those from your community who have benefitted from going to these private schools. If it has been occurring for such a long time and has come "into the light" why do your very own people continue to leave?
Why not condemn them also?
Some of the most notable and well respected families from your area have a family member or themselves who have benefitted from attending private schools. I dont understand this. They shouldnt be blind to what is happening now but it still is happening. What gives?
You are holding this coach to the fire but what about your own people? Wait does the community shun them when they send their child to private school, do you shame them? Are they corrupted for wanting to be a part of a corrupt system?
afan January 30, 2019 12:08 am
St. Louis is the reason college coaches came to Hawaii in droves. Now word is out that there is talent at other schools around the island. Just like that another kid from your neighborhood school off to college to represent Hawaii. Recruiting will continue in the ILH, it's their right as a school. Complaining doesn't change it. Unless you're planning on talking to every athlete and the parent about the many reasons to stay rooted…save it! Some kids go and some stay. Nothing anyone can do about it. I'm sure the same guys cheering against St. Louis is the same guys cheeering for Tua and Alabama on Saturdays!
@ILH thats what I've been saying too, if ur slamming the coach and ILH school for recruiting, then why u don't slam the individual student athlete as well as the parents who accept that scholarship offer. Slam the ILH school for doing that and then at the same time be fair the slam the kid and parents, b/c in your opinion, the didn't stand up for their community or have pride or blah blah blah blah blah. Then maybe I should do it, ur own kids and parents are weak for not being able to withstand temptation of scholarships and free money, what a bunch of losers yea notes? yea ???. Man good thing this process weeds out those families who are freaking donkeys in your community, who knows, maybe they will go on to play professional and just maybe they will come back after being successful and move their entire families out of your community and then u won't have to see them and have it b a reminder that they suck, and have no community pride!?!? @NOTES yup my family has NEVER had to deal with our family as a whole being ripped apart from the evil ILH b/c we don't see it as that, we see it as parents making the decision that they feel bests suits their kid and their future. We don't have this dumb victim mentality, we actually encourage growth and progress, whatever that may look like. We don't have this dumb "crab in a bucket" mentality. And lets just get real, its freakin high school sports, who the heck cares what families are cheering against or for. I mean seriously. The only one the seems to be torn and loosing much much much sleep and butt hurt is you apparently? I mean if we took a poll of your community, would it actually be that majority of them would say "yes we are the victims, yes we loose sleep over it, yes we need to go counseling over this, yes, my community is torn apart"?
@??? Am I crazy and living under a rock? If you were a business owner and on someones resume noticed the school they graduated from was your very own school you graduated from, would ur interested not peak b/c u both graduated from the same school? thats all Im' saying. Oh AND I don't know CEO's and business owners of large companies in hawaii who graduated from private school or anything. Thats none of my business right?
Wainakea January 30, 2019 5:55 pm
It's kinda nuts how people are so obsessed with recruiting…you don't go to school just to represent to your neighborhood. You go to school to become what you want to be. If a kid gets recruited and goes to an ILH school, good for them. It's sad from a public school standpoint, but to ban for the purposes above is putting the cart before the horse.
notes February 2, 2019 9:17 am
T103 the only one with a victim mentality is you. "my own community and community school is not good enough for myself or my family. if i don't get an ILH 'education' then nobody can ever succeed in life." it's the ultimate in victim mentality when you think your own home and community isn't good enough and then instead of working to build it up and make its football program successful, you join the people like "coach" cal who flat out admitted to recruiting in order to destroy the very communities you came from. tbh my heart actually hurts reading all these hurtful comments from people like you and all the other ILH apologists here
why would we expect any different though? the ILH has deep pockets and everyone has a price
T103 February 4, 2019 5:44 am
@notes k first off, let me get this squared away, my apologies if you are genuinely hurt over reading comments on here, not my intention. I think there is a bigger picture to look at in this whole thing.
It isn't all about football. Football isn't and shouldn't be "life" so to speak.
– Football is just part of the whole picture. If no one recruited and everyone stayed in their community, that doesn't guarantee good education. Private schools give better education then public school, and of course the parents pay through their nostrils for it. It's not like our public school system is rated so high. If we had a quality public school educational system, then I think a lot of parents would send their kids public school cuz it's free but parents will work 2-3 jobs just to put their kids through private school for better education.
– As a parent I wouldn't sacrifice sending my kid to the public school in my district just to "build the football program"
notes February 6, 2019 8:53 pm
if people like T103 invested in their own community schools instead of abandoning their own community when the ILH and "coach" cal comes calling with money, maybe the public school system would be better. it starts with kids having pride and staying home to play for their community. T103 can make all the excuses he wants but it's so sad to see how ILH greed has infected everyone. no matter how many kids you steal from our community, kahuku just reloads. we have the most state titles since state tournament became a thing and since HHSAA banned 19 year olds from playing for high school. people like T103 have hate in their hearts for the community schools they abandoned and want to see us fail but we will rise above the haters.
the state needs to actually investigate "coach" cal and the ILH recruiting agenda. you'd be surprised at how much corruption is in there. worse than the kealohas
t103 February 19, 2019 5:48 am
To think that parents who would send their kids to public schools in their communities would actually be able to "invest" in the public school system to improve it is laughable. If parents sent their children to their local schools, it wouldn't change anything b/c the amount of money it would take to actually improve the entire school system is millions upon millions. I mean for goodness sakes, it cost so much money to put AC in certain schools cuz it's so freaking hot, and thats not even every single school. We barely survive paying for basic necessities b/c the high cost of living. The sate that has the best school system in the U.S. somewhere on the mainland was only able to invest "community" money b/c they have the lottery system and they poured all that revenue back into the state school system and THAT is how they were able to have the #1 public school system I the nation.
And although I don't have kids, I have a niece and the school she is slated to go to for h.s. just passes students on to the next grade so the teachers don't have to deal with them for one more year if they flunk them and have to repeat a grade. A lot of violence amongst students, and a high population of immigrants who English isn't their first language so the entire curriculum for the entire school was changed to be a lot more simpler which left everyone else being "dumbed down" so to speak, and it's like they are in basic math, english, etc etc. The current students tell me that it's REALLY hard to fail that school and or get kicked out. What would you do in this situation? How would you be able to improve that school as you say? would you Send your child still yet? Or given the chance send your child to have a better opportunity for education??
And to say the feds need to investigate cal lee for recruiting
1. its not illegal
2. Those are lofty accusations to suggest that cal lee is doing more illegal things then abuse of power, selling drugs, using authority to cover up crimes, using other peoples computers to send out emails to mislead investigations, defrauding banks for loans, making up aliases to secure loans, stealing from children, stealing money from your grandmother.
I'm not a hater of public schools, I'm just being real.
Kakaako February 21, 2019 9:40 pm
Public schools get lucky if they recruit one kid every 10yrs. STL and Pun recruit 10 kids every year.
The recruiting is not the main issue. The issue is the merger/alliance and thinking the OIA is equal to the ILH so they should play in the same conference.
Damien, Iolani and Pac-5 chose to play in D2 and leave St. louis behind because of their excessive recruiting which led to blow outs and beat downs. STL is losing friends so they are forcing the OIA to play with them and combine regular seasons. STL is now beating down on the OIA with an average score of 49-7. Waianae, Farrington, Kapolei, Campbell, Mililani and Kahuku are all being used as rag dolls for the super team "STL" to beat up on.
STL needs Kahuku and Mililani to be good so that the games will be close but STL will always have access to more elite players state wide to ensure they are always "one up" on the OIA and win championships.
STL why dont you go play Gormon, Della salle or Mater Dei? Sked?
Lambert Chee March 4, 2019 10:11 am
I attended St. Louis on a true (academic) scholarship 50 years ago. For those in need, St. Louis offered 'tuition assistance'. I played no sports, but was able to attend an Ivy League school (on another scholarship) after graduation and medical school. Was I recruited, no, but I could not have attended without the scholarship (I did not know about the tuition assistance at that time.) So St. Louis is not all athletics, not even today.

redpajama_set_name: RedPajamaCommonCrawl


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text:
\section{Introduction}
In a graph $G$, a set $S\subseteq V(G)$ is \emph{independent} if $G[S]$ does not contain a copy of $K_2$. More generally for any integer $s$, a set $S \subseteq V(G)$ can be called $s$\emph{-independent} if $G[S]$ does not contain a copy of $K_s$. With this in mind, define the $s$\emph{-independence number} of $G$, denoted by $\alpha_s(G)$, to be the size of the largest $s$-independent set in $G$. The classical Ramsey number $R(t,u)$ can be defined in this language as the smallest integer~$n$ such that every graph of order~$n$ contains either a copy of $K_t$ or a 2-independent set of size $u$. In other words, $R(t,u)$ is the smallest integer $n$ such that
$$
u \leq \min \{\alpha_2(G): G \text{ is a } K_t \text{-free graph of order } n \}.
$$
Observe that if the right hand side of the above inequality is understood as a function of $n$ and $t$, then so is the classical Ramsey number.

A more general problem results by replacing the standard independence number by the $s$-independence number for some $2\le s<t$. Following this approach, in 1962 Erd{\H o}s and Rogers~\cite{ERog} introduced the function
$$
f_{s,t}(n)=\min \{\alpha_s(G): G \text{ is a } K_t \text{-free graph of order } n \}.
$$
\noindent The lower bound $k \le f_{s,t}(n)$ means that every $K_t$-free graph of order~$n$ contains a subset of $k$ vertices with no copy of~$K_s$. The upper bound $f_{s,t}(n) < \ell$ means that there exists a $K_t$-free graph of order~$n$ such that every subset of $\ell$ vertices contains a copy of~$K_s$.

The case $t = s+1$ has received the considerable attention over the last 50 years, in part due to the fact that it creates a general upper bound in the sense that for $t' > t$, we clearly have $f_{s,t'}(n) \leq f_{s,t}(n)$. A first nontrivial upper bound for $f_{s,s+1}(n)$ was established by Erd{\H o}s and Rogers~\cite{ERog}, which was subsequently addressed by Bollob\'as and Hind~\cite{BH}, Krivelevich~\cite{KR2, KR}, Alon and Krivelevich \cite{AK}, Dudek and R\"odl \cite{DR}, and most recently Wolfovitz~\cite{Wo}. The first nontrivial lower bound established by Bollob\'as and Hind~\cite{BH} was later slightly improved by Krivelevich~\cite{KR}. The most recent general bounds for $s\geq 3 $ were of the form:
\begin{equation}\label{eq:bound_s}
\Omega\left(\sqrt{\frac{n \log n }{\log\log n}} \right) = f_{s, s+1}(n) = O\big{(}n^{\frac{2}{3}} \big{)}.
\end{equation}
The precise lower bound of \eqref{eq:bound_s} was first explicitly stated by Dudek and Mubayi~\cite{DM}, and was based upon their observation that the result of Krivelevich \cite{KR} could be slightly strengthened by incorporating a result of Shearer \cite{SH}. This upper bound of \eqref{eq:bound_s} appears in \cite{DR}, where it was also conjectured that for all sufficiently large $s$ the upper bound could be improved to show that
\begin{equation}\label{conj:1}
f_{s,s+1}(n) = n^{\frac{1}{2}+o(1)}.
\end{equation}
Recently, Wolfovitz~\cite{Wo} showed that \eqref{conj:1} holds when $s=3$. In this paper, \eqref{conj:1} is proved for every $s \geq 3$, establishing an upper bound that is tight up to a polylogarithmic factor. Our proof builds upon the ideas in \cite{Wo}, \cite{DR}, \cite{KR2}, and \cite{KR}.

\begin{thm}\label{thm:s1}
For every $s \geq 3$ there is a constant $c=c(s)$ such that
$$
f_{s,s+1}(n) \le c (\log n)^{4s^2}\sqrt{n}.
$$
\end{thm}
\noindent

For the case $t = s+2$, it follows from a result of Sudakov~\cite{SU} (see also \cite{DR} for a simplified formula) that $f_{s,s+2}(n) = \Omega(n^{a_2})$, where $\frac{1}{a_2} = 2+\frac{2}{3s-4}$. On the other hand, clearly $f_{s,s+2}(n) \leq f_{s,s+1}(n)$. When $s \geq 4,$ we establish an improved upper bound that omits the logarithmic factor.
\begin{thm}\label{thm:s2}
For every $s \geq 4$ there is a constant $c=c(s)$ such that
$$
f_{s,s+2}(n) \le c\sqrt{n}.
$$
\end{thm}
\noindent
This establishes the following corollary which provides the best known bounds on $f_{s,t}(n)$ for $t < 2s$.
\begin{cor} \label{cor3}
For every $6 \leq s +2 \leq t$ there is a constant $c=c(s)$ such that
$$
f_{s,t}(n) \le f_{s,s+2}(n) \le c\sqrt{n}.
$$
\end{cor}
\noindent
When $t \geq 2s$, the upper bound $c(\log n)^{1/(s-1)} n^{s/(t+1)}$ of Krivelevich~\cite{KR2} remains best. For all values of $t>s+1$, the best lower bounds follow from a recursive formula defined by Sudakov~\cite{SU, SU2}. We will return to the these results concerning the general case in our concluding remarks. More related results are summarized in the survey~\cite{DR2}.

Now that our two main results have been stated, we turn our attention towards an old question of Erd\H{o}s~\cite{ER}, who asked if for fixed integers $s+1 < t$,
\begin{equation}\label{eq:erdos}
\lim_{n\to\infty} \frac{f_{s+1,t}(n)}{f_{s,t}(n)} = \infty.
\end{equation}
This central conjecture in the area is still wide open and asks for a rather precise estimation of $f_{s,t}(n)$. It is known due to Sudakov~\cite{SU2} that \eqref{eq:erdos} holds for
$$
(s,t)\in\{ (2,4), (2,5), (2,6), (2,7), (2,8), (3,6)\}.
$$
Observe that Theorem~\ref{thm:s2} together with the lower bound of \cite{KR} (and \cite{DR}) implies that for $s\ge 4$,
$$
 \frac{f_{s+1,s+2}(n)}{f_{s,s+2}(n)} \ge \frac{\Omega\left(\sqrt{\frac{n \log n }{\log\log n}} \right)}{O(\sqrt{n})}
= \Omega\left(\sqrt{\frac{\log n }{\log\log n}}\right)
\xrightarrow[n\to\infty]{}\infty.
$$
That is, \eqref{eq:erdos} holds for all pairs $(s,t) \in\{ (4,6), (5,7), (6,8), \dots \}$.

In what follows, consider $s$ to be an arbitrary fixed integer and $n$ sufficiently large, i.e. $n \geq n_0(s)$. We will show that there exists a $K_{s+1}$-free graph of order~$n$ such that every subset of $c (\log n)^{4s^2}\sqrt{n}$ vertices contains a copy of~$K_s$ and that there exists a $K_{s+2}$-free graph of order~$n$ such that every subset of $c \sqrt{n}$ vertices contains a copy of~$K_s$. Indeed, this establishes Theorems \ref{thm:s1} and \ref{thm:s2} as stated (for all $n$), since the constant factors can subsequently be inflated to accommodate the finitely many cases where $n \leq n_0$. For simplicity, we do not round numbers that are supposed to be integers either up or down; this is justified since these rounding errors are negligible to the asymptomatic calculations we will make.

In Section~\ref{sec:H}, we begin our construction by considering the random hypergraph $\mathbb{H}$ which is essentially the random hypergraph obtained from the affine plane by taking each hyperedge (line) with some uniform probability. We then use $\mathbb{H}$ in Section~\ref{sec:G} to construct a random graph $\mathbb{G}$ by replacing each hyperedge by a complete $s$-partite graph. In Section \ref{sec:proofs}, the proof of Theorem \ref{thm:s2} considers an induced subgraph of $\mathbb{G}$ whereas the proof of Theorem \ref{thm:s1} considers yet another random subgraph of $\mathbb{G}$ which is analyzed by way of the Local Lemma.

Below we will use the standard notation to denote the neighborhood and degree of $v \in G$ by $N_G(v)$ and $d_G(v)$ respectively.

\section{The Hypergraph $H$} \label{sec:H}

The {\em affine plane} of order $q$ is an incidence structure on a set of $q^2$ points and a set of $q^2+q$ lines such that: any two points lie on a unique line; every line contains~$q$ points; and every point lies on~$q+1$ lines. It is well known that affine planes exist for all prime power orders.
Clearly, an incidence structure can be viewed as a hypergraph with points corresponding to vertices and lines corresponding to hyperedges; we will use this terminology interchangeably.

In the affine plane, call lines $L$ and $L'$ \emph{parallel} if $L \cap L' = \emptyset$. In the affine plane there exist $q+1$ sets of $q$ pairwise parallel lines. (For more details see, e.g., \cite{CA}.) Let $(V, \mathcal{L})$ be the hypergraph obtained by removing a parallel class of $q$ lines from the affine plane or order~$q$. The following lemma establishes some properties of this graph.

\begin{lem} \label{lem:0}
For $q$ prime, the $q$-uniform, $q$-regular hypergraph $(V,\mathcal{L})$ of order $q^2$ satisfies:
\begin{enumerate}[label=(P\arabic*)]
\item\label{P:1} Any two vertices are contained in at most one hyperedge;
\item\label{P:2} For every $A \in {V \choose q}$, $|\{L \in \mathcal{L} : L \cap A \not = \emptyset \}| \geq \frac{q^2}{2}$.
\end{enumerate}
\end{lem}

\begin{proof}
By construction, $(V,\mathcal{L})$ is $q$-uniform, $q$-regular, and satisfies \ref{P:1}. Consider $A=\{v_1,v_2,\dots ,v_q\}$. Define $d_{+}(v_i) = |\{ L \in \mathcal{L} : L \cap \{v_1,v_2,\dots, v_{i}\} = \{v_i\} \}|$. Hence by property~\ref{P:1}, $d_+(v_i) \geq q-i+1$. We now compute
$$ |\{L \in \mathcal{L} : L \cap A \not = \emptyset \}| \geq \sum_{i=1}^q d_+(v_i) \geq {q+1 \choose 2} \geq \frac{q^2}{2}.$$
\end{proof}

The objective of this section is to establish the existence of a certain hypergraph $(V,\mathcal{L}') \subseteq (V,\mathcal{L})$ by considering a random sub-hypergraph of $(V,\mathcal{L})$. Preceding this, we introduce some terminology. Define
$$\mathcal{L}_A' = \{L \in \mathcal{L}' : L \cap A \not = \emptyset \}, \hspace{1cm} \text{ and } \hspace{1cm} \mathcal{L}'_{B, \gamma } = \{L \in \mathcal{L}' : |L \cap B| \geq \gamma \}.$$
Call $S \subseteq V$ $\mathcal{L}'$-\emph{complete} if every pair of points in $S$ is contained in some common line in $\mathcal{L}'$. Let $L(x,y)$ denote the unique line in $\mathcal{L}$ containing $x$ and $y$, provided such a line exists.

We will now distinguish 3 types of \emph{$\mathcal{L}'$-dangerous} subsets as depicted in Figure \ref{fig:ks2}. The first two types have 5 vertices $\{v_1,v_2,v_3,v_4,x\}$ and third type has 6 vertices $\{v_1,v_2,v_3,v_4,y,z\}$. All 3 types of dangerous sets must be $\mathcal{L}'$-complete and have 4 points $\{v_1,v_2,v_3,v_4\}$ in general position. Additionally we specify:
\begin{description}
  \item[Type 1 $\mathcal{L}'$-dangerous] \hfill \\
  The points $\{v_1,v_2,v_3,v_4,x\}$ are in general position.
  \item[Type 2 $\mathcal{L}'$-dangerous] \hfill \\
  The point $x$ is contained in precisely one of the 6 lines $L(v_i,v_j)$ for $1 \leq i < j \leq 4$. Up to relabeling, say $x \in L(v_2,v_3)$.
  \item[Type 3 $\mathcal{L}'$-dangerous] \hfill \\
  The points $y$ and $z$ are each contained in exactly two of the lines $L(v_i,v_j)$ for $1 \leq i < j \leq 4$. Up to relabeling, say $y \in L(v_1,v_3) \cap L(v_2,v_4)$ and $z \in L(v_1,v_2) \cap L(v_3,v_4)$.
\end{description}
All concepts above were defined relative to the subset $\mathcal{L}' \subseteq \mathcal{L}$. Obviously we can define the concepts $\mathcal{L}$-complete, $\mathcal{L}$-dangerous, $\mathcal{L}_A$, and $\mathcal{L}_{B, \gamma }$ related to the set $\mathcal{L}$ analogously.

\begin{figure}
\begin{center}
\subfigure[Type 1]{\includegraphics[scale=0.4]{d1.pdf}}
\qquad
\subfigure[Type 2]{\includegraphics[scale=0.4]{d2.pdf}}
\qquad
\subfigure[Type 3]{\includegraphics[scale=0.4]{d3.pdf}}
\caption{Types of dangerous sets.}
\label{fig:ks2}
\end{center}
\end{figure}

We are now ready to state the main result of this section.

\begin{lem} \label{lem:1}
Let $q$ be a sufficiently large prime and $\alpha = (\log q)^2$.
Then, there exists a $q$-uniform hypergraph $H=(V,\mathcal{L}')$ of order $q^2$ such that:
\begin{enumerate}[label=(H\arabic*)]
\item\label{H:0} Any two vertices are contained in at most one hyperedge;
\item\label{H:3} For every $v \in V$, $d_H(v) \leq 2 \alpha$;
\item\label{H:4} $|\mathcal{D}| \leq 2 \alpha^8 q$, where $\mathcal{D}$ is the set of $\mathcal{L}'$-dangerous subsets;
\item\label{H:1} For every $A \in {V \choose q}$, $|\mathcal{L}_A'|\geq \frac{\alpha q}{4}$;
\item\label{H:5} For every integer $1 \leq \gamma \leq \frac{q}{16}$ and every $B \in {V \choose 16 \gamma q}$, $|\mathcal{L}'_{B, \gamma }| \geq \frac{\alpha q}{8}$.
\end{enumerate}
\end{lem}

Before proving the above lemma, we state a basic form of the Chernoff bound (as appearing in Corollary 2.3 of \cite{JLR}) and mention what we will refer to as the union bound.

\begin{c1}
If $X \sim \text{Bi}(n,p)$ and $0 < \varepsilon \leq \frac{3}{2}$, then
$$\Pr \Big( |X - E(X)| \geq \varepsilon \cdot E(X) \Big) \leq 2 \exp \left\{ -\frac{E(X) \varepsilon^2 }{3} \right\} .$$
\end{c1}

\begin{ub}
If $E_i$ are events, then
$$\Pr \Big( \bigcup_{i=1}^k E_i \Big) \leq k \cdot \max \{\Pr(E_i): i \in [k]\}.$$
\end{ub}

\begin{proof}[Proof of Lemma~\ref{lem:1}] Take $(V,\mathcal{L})$ to be a hypergraph established by Lemma \ref{lem:0}. Let $\mathbb{H}=(V,\mathcal{L}')$ be a random sub-hypergraph of $(V,\mathcal{L})$ where every line in $\mathcal{L}$ is taken independently with probability
$$
\frac{\alpha}{q}=\frac{(\log q)^2}{q}.
$$
Since $\mathbb{H}$ is a subgraph of $(V, \mathcal{L})$ any two vertices are in at most one line, so $\mathbb{H}$ always satisfies \ref{H:0}. We will show $\mathbb{H}$ fails to satisfy \ref{H:3} and \ref{H:1} with probability at most $o(1)$ and that $\mathbb{H}$ fails to satisfy \ref{H:4} with probability at most $\frac{1}{2}$. Together this implies $\mathbb{H}$ satisfies \ref{H:0}-\ref{H:1} with probability at least $1-\frac{1}{2}-o(1)$, establishing the existence of a hypergraph $H$ that satisfies \ref{H:0}-\ref{H:1}. Finally, we use a counting argument to show that any such $H$ necessarily satisfies \ref{H:5}.

\emph{\ref{H:3}:} We first show that the probability that there exists a vertex of degree greater than $2 \alpha$ is $o(1)$. Observe for fixed $v \in \mathbb{H}$, $d_{\mathbb{H}}(v) \sim Bi(q, \frac{\alpha}{q})$ and $E(d_{\mathbb{H}}(v))= \alpha.$ So by the Chernoff bound with $\varepsilon = 1$,
$$ \Pr \Big( d_{\mathbb{H}}(v) \geq 2 \alpha \Big) \leq \Pr \Big( |d_{\mathbb{H}}(v)-\alpha| \geq \alpha \Big) \leq 2  \exp \left\{ -\frac{\alpha}{3} \right\}. $$
Thus by the union bound the probability that there exists some $v \in V$ with $d_{\mathbb{H}}(v) \geq 2 \alpha$ is at most
$$q^2 \cdot 2  \exp \left\{-\frac{\alpha}{3}\right\} = 2 \exp \left\{2 \log q-\frac{(\log q)^2}{3}\right\} = o(1).$$

\emph{\ref{H:4}:} In order to show $|\mathcal{D}| > 4 \alpha^8 q$ with probability at most $\frac{1}{2}$, we begin by counting the number of $\mathcal{L}$-dangerous subsets of each type. Clearly the number of Type 1 $\mathcal{L}$-dangerous subsets is at most ${q^2 \choose 5}$. To count the number of Type 2 $\mathcal{L}$-dangerous subsets, first choose $\{v_1,v_2,v_3,v_4\}$ then $x$, observing $x$ must lie on one of the 6 lines which each have at most $q$ vertices. Thus there are at most ${q^2 \choose 4} (6q)$ configurations of this type. To count the number of Type 3 $\mathcal{L}$-dangerous subsets, observe the lines $L(v_i,v_j)$ for $1 \leq i < j \leq 6$ intersect at at most 3 points other than $v_1,v_2,v_3,v_4$. Hence there are at most ${q^2 \choose 4} {3 \choose 2}$ subsets of this type in $\mathcal{L}$.

Since $\mathcal{L}$-dangerous subsets of Type 1, Type 2, and Type 3 have 10, 8, and 7 lines respectively, an $\mathcal{L}$-dangerous subset of each type will be $\mathcal{L}'$-dangerous with respective probabilities $\left(\frac{\alpha}{q}\right)^{10}, \left(\frac{\alpha}{q}\right)^8$, and $\left(\frac{\alpha}{q}\right)^7$. By the linearity of expectation, we now compute
$$ E(|\mathcal{D}|) \leq {q^2 \choose 5} \cdot \left( \frac{\alpha}{q} \right)^{10}+ {q^2 \choose 4} (6q) \cdot \left( \frac{\alpha}{q} \right)^{8} + {q^2 \choose 4} {3 \choose 2} \cdot \left( \frac{\alpha}{q} \right)^{7} \le \alpha^{10} + \frac{q \alpha^8}{4} + \frac{q \alpha^7}{8}  \leq q \alpha^8. $$

Thus, the Markov inequality yields,
$$\Pr \Big( |\mathcal{D}| \ge 2 \alpha^8 q \Big) \le \Pr \Big( |\mathcal{D}| \ge 2 E ( |\mathcal{D}|) \Big) \le \frac{1}{2}.$$

\emph{\ref{H:1}:} We will now prove that the probability that there exists $A \in {V \choose q}$ such that $|\mathcal{L}_A'| < \frac{\alpha q}{4}$ is $o(1)$. Begin by considering any fixed $A \in {V \choose q}$. Then by Lemma \ref{lem:0}, $|\mathcal{L}_A| \geq \frac{q^2}{2}$, so we may fix $X \subseteq \mathcal{L}_A$ with $|X|=\frac{q^2}{2}$. Let $X'= X \cap \mathcal{L}'$. Since each line in $X$ appears in $X'$ independently with probability $\frac{\alpha}{q}$, $|X'| \sim Bi(\frac{q^2}{2},\frac{\alpha}{q})$ and $E(|X'|)=\frac{\alpha q}{2}$. Hence by the Chernoff bound with $\varepsilon=\frac{1}{2}$,
$$\Pr \Big( |\mathcal{L}_A'| < \frac{\alpha q}{4} \Big) \leq \Pr  \Big( |X'|< \frac{\alpha q}{4} \Big) \leq \Pr \Big( \Big| |X'|-\frac{\alpha q}{2} \Big|  \geq  \frac{\alpha q}{4} \Big) \leq 2\exp\left\{ - \frac{\alpha q}{24} \right\}.$$
Consequently by the union bound, the probability that there exits some $A \subseteq V$, $|A|=q$, with $|\mathcal{L}_A'| < \frac{\alpha q}{4} $ is at most

$$ {q^2 \choose q} \cdot 2\exp\left\{- \frac{\alpha q}{24} \right\} \leq q^{2q} \cdot 2\exp\left\{ - \frac{(\log q)^2 q}{24} \right\} = 2 \exp\left\{2q \log q - \frac{q(\log q)^2}{24} \right\}=o(1).$$

\emph{\ref{H:5}:} Finally, we will establish the following deterministic property: If $H$ satisfies \ref{H:3} and \ref{H:1}, then $H$ also satisfies \ref{H:5}.

Consider arbitrary fixed $0 \leq \gamma \leq \frac{q}{16}$ and $B \in {V \choose 16 \gamma q}$. Let $B=B_1 \cup B_2 \cup \dots \cup B_{16 \gamma}$ be a partition of $B$ into $16 \gamma$ sets of size $q$. Consider the auxiliary bipartite graph \emph{Aux} with bipartition $\{B_1, B_2, \dots, B_{16 \gamma}\} \cup \mathcal{L}'$. Join $B_i$ to $L \in \mathcal{L}'$ if $B_i \cap L \not = \emptyset$. By property \ref{H:1} $d_{Aux}(B_i) \geq \frac{\alpha q}{4}$ for all $i \in [16 \gamma]$, and thus the number of edges in \emph{Aux} satisfies
\begin{equation}\label{eq:H5a}
|e(Aux)| \geq \frac{\alpha q}{4} |\{B_1, B_2, \dots, B_{16 \gamma}\}|=4 \alpha q \gamma.
\end{equation}
 On the other hand, clearly $d_{Aux}(L) \leq |\{B_1, B_2, \dots, B_{16 \gamma}\}| = 16 \gamma$ for all $L \in \mathcal{L}'$ and by definition $d_{Aux}(L') \leq \gamma$ for all $L' \in \{ \mathcal{L}' \setminus \mathcal{L}_{B, \gamma }' \}$. Also keeping in mind that by \ref{H:3} \newline
$| \mathcal{L}' \setminus \mathcal{L}_{B, \gamma }' | \leq |\mathcal{L}'| = \sum_{v \in V} \frac{d_H(v)}{q} \leq q^2\frac{2 \alpha}{q}=2 \alpha q$, we compute
\begin{equation}\label{eq:H5b}
|e(Aux)|  \leq |\mathcal{L}_{B, \gamma }'| \cdot 16 \gamma + |\{ \mathcal{L}' \setminus \mathcal{L}_{B, \gamma }' \}| \cdot \gamma \leq |\mathcal{L}_{B, \gamma }'| \cdot 16 \gamma + 2 \alpha q \cdot \gamma.
\end{equation}
Comparing (\ref{eq:H5a}) and (\ref{eq:H5b}), we obtain
$$4 \alpha q \gamma \leq |e(Aux)|  \leq |\mathcal{L}_{B, \gamma }'| \cdot 16 \gamma + 2 \alpha q \gamma,$$
which yields $|\mathcal{L}'_{B, \gamma }| \geq \frac{\alpha q}{8}$.
\end{proof}

\section{The Graph $G$}\label{sec:G}
Based upon the hypergraph $H$ established in the previous section, we will construct a graph $G$ with the following properties.
\begin{lem} \label{lem:3}
Let $q$ be a sufficiently large prime, $\alpha = (\log q)^2$,  $\beta=(\log q)^{4s^2}$, and $s\ge 3$. Then, there exists a graph $G=(V,E)$ of order $q^2$ such that:
\begin{enumerate}[label=(G\arabic*)]
\item\label{G:1} For every $C \in {V \choose 16 s q}$, $G[C]$ contains a copy of $K_s$;
\item\label{G:2} For every $U \in {V \choose 64 s \beta q}$, $G[U]$ contains $\frac{\alpha \beta^2 q}{16}$ edge disjoint copies of~$K_s$;
\item\label{G:3} Every edge $xy \in E$ is in at most $6^s \alpha^{2s-2}$ copies of $K_{s+1}$;
\item\label{G:4} If $s\ge 4$, then $G$ can be made $K_{s+2}$ free by removing $2 \alpha^8 q $ vertices.
\end{enumerate}
\end{lem}

\begin{proof}  Fix a hypergraph $H=(V,\mathcal{L}')$ as established by Lemma~\ref{lem:1}. Construct the random graph $\mathbb{G}$ as follows. For every $L\in \mathcal{L}'$, let $\chi_L : L \to [s]$ be a random partition of the vertices of $L$ into $s$ classes, where for every $v \in L$, a class $\chi_L(v) \in [s]$ is assigned uniformly and independently at random. Then, let $xy \in E$ if $\{x,y\} \subseteq L$ for some $L \in \mathcal{L}'$ and $\chi_L(x) \not = \chi_L(y)$. Thus for every $L \in \mathcal{L}'$, $\mathbb{G}[L]$ is a complete $s$-partite graph with vertex partition $L = \chi^{-1}_L (1) \cup \chi^{-1}_L (2) \cup \dots \cup \chi^{-1}_L (s)$ (where the classes need not have the same size and the unlikely event that a class is empty is permitted). Observe that not only are $G_{H'}[L]$ and $G_{H'}[L']$ are edge disjoint for distinct $L, L' \in \mathcal{L}'$, but also that the partitions for $L$ and $L'$ were determined independently.

We will show $\mathbb{G}$ does not satisfy \ref{G:1} and \ref{G:2} with probability at most $o(1)$ and that $\mathbb{G}$ always satisfies \ref{G:3} and \ref{G:4}. Hence the probability that $\mathbb{G}$ satisfies properties \ref{G:1}-\ref{G:4} is at least $1-o(1)$, implying the existence of a graph $G$ described in the lemma.

\emph{\ref{G:1}:} Consider any $C \in {V \choose 16sq}$. We will bound the probability that $G[C] \not \supset K_s$. By \ref{H:5} with $\gamma = s$, the set of lines $\mathcal{L}'_{C,s}$ that intersect $C$ in at least $s$ vertices has cardinality $|\mathcal{L}'_{C,s}| \geq \frac{\alpha q}{8}$. For each $L \in \mathcal{L}'_{C,s}$, let $X_L$ be the event $K_s \not \subseteq \mathbb{G}[L \cap C]$. Since $|L \cap C| \geq s$ for all $L \in \mathcal{L}'_{C,s}$, $\Pr (X_L) \leq 1- \frac{s!}{s^s}$. By independence,
$$\Pr \Big( K_s \not \in \mathbb{G}[C] \Big) \leq \Pr \Big( \bigcap_{L \in \mathcal{L}'_{C,s}} X_L \Big) \leq {\left( 1- \frac{s!}{s^s} \right) }^{|\mathcal{L}'_{C,s}|} \leq {\left( 1- \frac{s!}{s^s} \right) }^\frac{\alpha q}{8}\leq \exp { \left\{ - \frac{s!}{s^s}\frac{\alpha q }{8} \right\} }.$$
So by the union bound, the probability that there exists a subset of $16 s q$ vertices in $\mathbb{G}$ that contains no $K_s$ is at most
$$
{q^2 \choose 16 s q} \exp { \left\{ - \frac{s!}{s^s}\frac{\alpha q }{8} \right\} }
\le  q^{16 s q} \exp { \left\{ - \frac{s!}{s^s}\frac{\alpha q }{8} \right\} }
= \exp\left\{16 s q \log q - \frac{s! q (\log q)^2}{8s^s} \right\} = o(1).
$$

\emph{\ref{G:2}:} For arbitrary $U \in {V \choose 64 s \beta q}$, we will bound the probability that $G[U]$ does not contain $\frac{\alpha \beta^2 q}{16}$ edge disjoint copies of $K_s$. By \ref{H:5} with $\gamma = 4 s \beta$, we may fix a subset $\mathcal{Z}_U \subseteq \mathcal{L}_{U,4 s \beta}'$ of exactly $\frac{\alpha q}{8}$ lines with the property that each line has intersection at least $4 s \beta$ with $U$. We will consider the lines in $\mathcal{Z}_U$ that contain the complete balanced $s$-partite graph on $2s \beta$ vertices, which we denote by $K_{2\beta,\dots,2\beta}$. Define $\mathcal{Z}_U' = \{L \in \mathcal{Z}_U : K_{2\beta,\dots,2\beta} \subseteq \mathbb{G}[L \cap U] \}$. The graph $K_{2\beta,\dots,2\beta}$ certainly contains at least $\beta^2$ edge disjoint $K_s$ (Since we may choose a prime $\beta \leq p \leq 2\beta$ and it follows from \cite{ACD} that we may then decompose $K_{p,\dots,p}$ into $p^2$ edge disjoint copies of $K_s$; this suffices for our purposes, but stronger results are know). Thus if we show $|\mathcal{Z}_U'| \geq \frac{\alpha q}{16}$ it will imply that $\mathbb{G}[U]$ contains at least $|\mathcal{Z}_U'| \cdot \beta^2 \geq \frac{\alpha \beta^2 q}{16}$ edge disjoint copies of $K_s$.

For $L \in \mathcal{Z}_U$, let $Y_L$ be the event that $L \not \in \mathcal{Z}_U'$ and fix $L_{4 s \beta} \subseteq L \cap U, |L_{4 s \beta}|=4s \beta$. Now $Y_L$ will occur only if $|\chi_{L}^{-1}(i) \cap L_{4 s \beta}| < 2 \beta$ for some $i \in [s]$. Defining $X_i = |\chi_{L}^{-1}(i) \cap L_{4 s \beta}|$, observe $X_i  \sim Bi(4s\beta, \frac{1}{s})$ and $E(X_i) = 4 \beta$. Chernoff's inequality reveals
$$\Pr \Big( X_i < 2 \beta \Big) \leq \Pr \Big( |X_i - E(X_i)| \geq \frac{E(X_i)}{2} \Big) \leq 2 \exp\left\{-\frac{4 \beta }{12}\right\}=2 \exp\left\{-\frac{\beta }{3}\right\}.$$
By the union bound, $\Pr ( Y_L ) \leq \Pr \Big( \bigcup_{i \in s} (X_i \leq 2 \beta)  \Big) \leq s \cdot 2 \exp\left\{-\frac{ \beta }{3}\right\}.$

By independence, the probability that $Y_L$ occurs for at least $\frac{\alpha q}{16}=\frac{|\mathcal{Z}_U|}{2}$ of the lines in $\mathcal{Z}_U$ is at most

$$\binom{|\mathcal{Z}_U|}{\frac{|\mathcal{Z}_U|}{2}} \left( 2s \exp\left\{-\frac{ \beta}{3}\right\} \right)^{|\mathcal{Z}_U| / 2}
\le 4^{|\mathcal{Z}_U| / 2} \left( 2s \exp\left\{-\frac{ \beta}{3}\right\} \right)^{|\mathcal{Z}_U| / 2}  = \left( 8s \exp\left\{-\frac{ \beta}{3}\right\} \right)^{\frac{\alpha q}{16}}.$$

That is, we have shown $|\mathcal{Z}_U'| < \frac{\alpha q}{16}$ with probability at most $\left( 8s \exp\left\{-\frac{\beta}{3}\right\} \right)^{\frac{\alpha q}{16}}$ for fixed~$U$. Thus by the union bound, the probability that there exits some $U \subseteq V$ with $|U|= 64 s \beta q$ such that $|\mathcal{Z}_U'| < \frac{\alpha q}{16}$  is at most

\begin{align*}
{{q^2} \choose {64 s \beta q}} \left( 8s \exp\left\{ - \frac{\beta}{3}  \right\} \right)^{\frac{\alpha q}{16}}
&\le q^{64s \beta q } ( 8s )^{\frac{\alpha q}{16}} \left( \exp\left\{ - \frac{\beta}{3}  \right\} \right)^{\frac{\alpha q}{16}} \\
&\leq  \exp \left\{64 s \beta q \log q + \frac{\alpha q}{16} \log(8s) -  \frac{\alpha \beta q }{48} \right\}=o(1).
\end{align*}

\emph{\ref{G:3}:} For any $xy \in E$, we will show the number of copies of $K_{s+1}$ that contain $xy$ is at most $6^s \alpha^{2s-2}$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{ks1.pdf}
\caption{Counting $K_{s+1}$ in $\mathbb{G}$ that contains a fixed edge $xy$ by considering lines in $H$.}
\label{fig:ks1}
\end{figure}
Let $L \in \mathcal{L}'$ be the unique line such that $\{x,y\} \subseteq L$ as depicted in Figure~\ref{fig:ks1}. Let $N = (N_H(x) \cap N_H(y)) \setminus L$ be the set of all vertices not on $L$ that are collinear with both $x$ and $y$. Since $d_H(x) , d_H(y) \leq  2\alpha$ by \ref{H:3}, we infer that $|N| \leq 4{\alpha}^2$. Because $K_{s+1} \not \subseteq G[L]$, if a $K_{s+1}$ is to contain $x$ and $y$ it must contain at least one vertex $v \in N$. There are at most $|N| \le 4{\alpha}^2$ choices for this vertex $v$. Once $v$ has been chosen, each of the remaining $s-2$ vertices of the $K_{s+1}$ must lie in $N$ or in $L \cap N_H(v)$. Since $|N|+ |L \cap N_H(v) | \leq 4{\alpha}^2+2 \alpha$, the number of $K_{s+1}$ containing the edge $xy$ is at most ${4 \alpha}^2 (4{\alpha}^2+2\alpha)^{s-2} \leq 6^s \alpha^{2s-2}$.

\emph{\ref{G:4}:} We will finally show that if $s \geq 4$, $G$ can be made $K_{s+2}$ free be removing at most $2 \alpha^8 q $ vertices. By \ref{H:4}, all $\mathcal{L}'$-dangerous sets can be destroyed by removing $2 \alpha^8 q$ vertices, so it suffices to shown that every $K_{s+2}$ in $\mathbb{G}$ contains a $\mathcal{L}'$-dangerous subset.

Let $K$ be any copy of $K_{s+2}$ in $\mathbb{G}$. By assumption $s \geq 4$, so $K$ must have at least 6 vertices, which clearly form a $\mathcal{L}'$-complete set.

We first show that $K$ contains 4 vertices in general position. Suppose otherwise. Then there is some line $L \in \mathcal{L}'$ that contains 3 vertices $\{p_1,p_2,p_3\}$ of $K$. Since $K_{s+1} \not \subseteq \mathbb{G}[L]$, there must exist two vertices $a$ and $b$ in $K \setminus L$. Observe $\{a,b\}$ and any 2 vertices in $\{p_1,p_2,p_3\} \setminus L(a,b)$ are in general position.

Now fix 4 vertices $\{v_1,v_2,v_3,v_4\}$ of $K$ that are in general position and let $u_1,u_2$ be any two other vertices of $K$. Three cases are now considered. If either $u_1$ or $u_2$ do not lie on any of the 6 lines $L(v_i,v_j)$ for $1 \leq i < j \leq 4$, then there is a $\mathcal{L}'$-dangerous subset of Type~1. If either $u_1$ or $u_2$ lie on exactly one line in $L(v_i,v_j)$ for $1 \leq i < j \leq 4$, then there is a $\mathcal{L}'$-dangerous subset of Type~2. In the remaining case where both $u_1$ and $u_2$ each lie on at least 2 lines in $L(v_i,v_j)$ for $1 \leq i < j \leq 4$, then there is a $\mathcal{L}'$-dangerous subset of Type~3.
\end{proof}

\section{Proof of Theorem \ref{thm:s1} and \ref{thm:s2}}\label{sec:proofs}

Consider any sufficiently large integer $n$ and $s \geq 3$. By Bertrand's postulate, we can find a prime $q$ such that $4n \leq q^2 \leq 16n$. Fix a graph $G$ procured by Lemma \ref{lem:3} of order $q^2$ and as before take
$$
\alpha = (\log q)^{2} \hspace{1cm} \text{and} \hspace{1cm} \beta = (\log q)^{4s^2}.
$$
Theorem~\ref{thm:s1} and Theorem~\ref{thm:s2} are now proved by considering different subgraphs of $G$ of order $n$.

\begin{proof}[Proof of Theorem \ref{thm:s2}]
Consider the case where $s \geq 4$. To prove the theorem, we will show there exists a $K_{s+2}$-free induced subgraph of $G$ of order $n$ with the property that every subset of order $64 s \sqrt{n}$ contains a copy of $K_s$.

By \ref{G:1}, every set of size $16  s  q $ in $G$ contains $K_s$, so certainly every subset of size $64 s \sqrt{n} \geq 16  s  q$ in any induced subgraph of $G$ must also contain a copy of $K_s$.  Thus it will suffice to show that there is a $K_{s+2}$-free subset of $G$ of order $n$. But by \ref{G:4}, we know that there is a set $R \subseteq V(G)$ of size $|R|=2 \alpha^8 q \leq n$ such that $G [V \setminus R]$ will be $K_{s+2}$-free. Finally since $|V \setminus R | \geq 4n - n \geq n$, the induced graph of $G$ on any $n$ vertices in $V \setminus R$ will have the desired properties.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:s1}]
For $s \geq 3$, we will concentrate on constructing a $K_{s+1}$-free graph $G'$ on $q^2$ vertices with the property that every subset of size $64s \beta q$ vertices contains a copy of $K_s$. Since $\log(4n)\le 2\log n $,
$$
64s \beta q = 64 s (\log q)^{4s^2} q \leq
64 s (\log 4n)^{4s^2} 4n \leq
2^{4s^2+8} (\log n)^{4s^2} n,
$$
and so any induced subgraph of $G'$ of order $n$ will also be $K_{s+1}$-free and have the property that every set of order $2^{4s^2+8} (\log n)^{4s^2} n$ contains a copy of $K_s$, exactly as desired.

Let $G'$ be a random subgraph of $G$ where each edge is taken with probability
$$
\frac{1}{\gamma}, \quad \text{where}\ \gamma = (\log q)^{8}.
$$
For a set $S \in {V(G) \choose s+1}$ that spans a copy of $K_{s+1}$ in $G$, let $A_S$ to be the event that all the edges of $S$ are in $G'$. Hence, $\bigcap{\overline{A_S}} $ means that $K_{s+1} \not \subseteq G'$. For a set $U \in {V(G) \choose 64 s \beta q}$ let $\mathcal{K}_U$ be a (fixed) set of
$$
m=\frac{1}{16}\alpha\beta^2q
$$
edge disjoint copies $K_s$ contained in $U$, which are known to exist by \ref{G:2}. Define $B_U$ to be the event that none of the $m$ edge disjoint $K_s$ appear in $G'$. Hence, $ \bigcap \overline{B_U} $ implies that for every $U \in {V(G) \choose 64 s \beta q}$ one of the disjoint copies of $K_s$ in $G[U]$ appears in $G'$. It will suffice to show that the probability that $\left(\bigcap{\overline{A_S}} \right) \cap  \left( \bigcap \overline{B_U} \right)$ occurs is nonzero. In order to show this, we apply the Local Lemma (see, e.g., Lemma 5.1.1 in~\cite{AS}).

\begin{lovasz}
Let $E_1, E_2,\dots, E_k$ be events in an arbitrary probability space. A directed graph $D$ on the set of vertices $\{1,2,\dots,k\}$ is called a {\em dependency digraph} for the events $E_1, E_2,\dots, E_k$ if for each $i$, $1\le i \le k$, the event $E_i$ is mutually independent of all the events $\{E_j : (i, j) \not \in D\}$. Suppose that $D$ is a dependency digraph for the above
events and suppose there are real numbers $z_1,\dots, z_k$ such that $0 \le z_i < 1$ and
$Pr(E_i) \le z_i \prod_{(i,j)\in D} (1-z_j)$ for all $1\le i\le k$. Then,
$\Pr\left(\bigcap_{i=1}^k {\overline{E_i}} \right) > 0$.
\end{lovasz}

Let $D$ be a dependency graph that corresponds to all events $A_S$ and $B_U$.
Observe that $A_S$ depends only on the ${s+1 \choose 2}$ edges in $S$ and $B_U$ depends only on the $m {s \choose 2}$ edges of the $K_s$ in $\mathcal{K}_U$. Also, observe that the number of events of the type $B_U$ is ${q^2 \choose 64 s \beta q} \leq q^{64 s \beta q}$. Thus by~\ref{G:3}, a fixed event $A_S$ depends on at most
$$
d_{AA} = {s+1 \choose 2}  6^s \alpha^{2s-2}
$$
other events $A_{S'}$ and at most
$$
d_{AB} = q^{64 s \beta q}
$$
events $B_U$. Similarly, a fixed event $B_U$ depends on at most
$$
d_{BA} = m {s \choose 2}  6^s \alpha^{2s-2}
$$
events $A_S$ and at most
$$
d_{BB} = q^{64 s \beta q}
$$
other events $B_{U'}$. Let
$$
x= \frac{1}{\alpha^{2s^2}}\quad \text{and} \quad y= \frac{1}{(\log q)^{4s^2} q^{64 s \beta q}}.
$$
To finish the proof, due to the Local Lemma it suffices to show that
\begin{equation}\label{eq:lllx}
{\left( \frac{1}{\gamma} \right)}^{s+1 \choose 2} \leq x(1-x)^{d_{AA}}(1-y)^{d_{AB}},
\end{equation}
\begin{equation}\label{eq:llly}
\left(1-\left( \frac{1}{\gamma} \right)^{s \choose 2}\right)^{m} \leq y (1-x)^{d_{BA}}(1-y)^{d_{BB}}.
\end{equation}

First we show that \eqref{eq:lllx} holds. Using the fact that $e^{-2x} \leq 1-x$ for $x$ sufficiently small (observe that $x \to 0$ with $q \to \infty$), a sufficient condition for \eqref{eq:lllx} will be
$$
{\left( \frac{1}{\gamma} \right)}^{s+1 \choose 2} \leq x\, e^{-2xd_{AA}}\, e^{-2yd_{AB}},
$$
and equivalently,
$$
{s+1 \choose 2} \log \left( \gamma \right) \geq \log \left( \frac{1}{x} \right)+2xd_{AA}+2yd_{AB}.
$$
The latter immediately follows from the following three inequalities (which can be easily verified):
\begin{align*}
&\frac{2s^2}{2s^2+2s}{s+1 \choose 2} \log \left( \gamma \right) \geq \log \left( \frac{1}{x} \right),\\
&\frac{s}{2s^2+2s} {s+1 \choose 2} \log \left( \gamma \right) \geq 2xd_{AA},\\
&\frac{s}{2s^2+2s} {s+1 \choose 2} \log \left( \gamma \right) \geq 2yd_{AB}.
\end{align*}

Similarly, using the facts that $e^{-2y} \leq 1-y$ for $y$ sufficiently small and that  $1-\left( \frac{1}{\gamma} \right)^{s \choose 2} \leq e^{-\left( \frac{1}{\gamma} \right)^{s \choose 2}}$, \eqref{eq:llly} will be satisfied if
$$
e^{-m {\left( \frac{1}{\gamma} \right)}^{s \choose 2}}  \le  y\, e^{-2xd_{BA}}\, e^{-2yd_{BB}},
$$
and equivalently,
$$
m {\left( \frac{1}{\gamma} \right)}^{s \choose 2}  \geq  \log {\left( \frac{1}{y} \right) } +2xd_{BA} +2yd_{BB}.
$$
As before the latter will follow from the following easy to check inequalities:
\begin{align*}
&\frac{1}{3} m {\left( \frac{1}{\gamma} \right)}^{s \choose 2}  \geq \log  {\left( \frac{1}{y} \right) },\\
&\frac{1}{3} m {\left( \frac{1}{\gamma} \right)}^{s \choose 2}  \geq  2xd_{BA},\\
&\frac{1}{3} m {\left( \frac{1}{\gamma} \right)}^{s \choose 2}  \geq  2yd_{BB}.
\end{align*}
This completes the proof of Theorem~\ref{thm:s1}.
\end{proof}

\section{Concluding Remarks}

We close this paper by discussing how the asymptotic behavior of $f_{s,t}(n)$ changes for different values of $3 \le s <t$.

If the difference between $s$ and $t$ is fixed, we make the following observation based upon the lower bound in Sudakov~\cite{SU} (and Fact 3.5 in~\cite{DR}) and Corollary \ref{cor3}.
\begin{obs}
For any $\varepsilon > 0$ and an integer $k\ge 2$ there is a constant $s_0=s_0(k, \varepsilon)$ such that for all $s\ge s_0$,
$$
\Omega\big{(}n^{\frac{1}{2} - \varepsilon}\big{)} = f_{s,s+k}(n) =
O(\sqrt{n}).
$$
\end{obs}
\noindent
In view of this observation and Theorem~\ref{thm:s2} we ask the following.
\begin{que}
For any $s\ge 3$, is $f_{s,s+2}(n) =  o(\sqrt{n})$?
\end{que}

Another interesting question results from fixing that ratio between $s$ and $t$. The following is based upon \cite{SU} and \cite{KR2} respectively.
\begin{obs}\label{obs:2}
For any $\varepsilon > 0$ and $\lambda \geq 2$ there is a constant $ s_0=s_0(\lambda, \varepsilon)$ such that for all $s\ge s_0$,
$$\Omega\big{(}n^{\frac{1}{2\lambda} - \varepsilon}\big{)} = f_{s,\lfloor \lambda s \rfloor}(n) = O\big{(}n^{\frac{1}{\lambda}}\big{)}.$$
\end{obs}
\noindent
In particular, when $\lambda =3$, we see $\Omega(n^{1/6 - \varepsilon}) = f_{s,\lfloor \lambda s \rfloor}(n) = O(n^{1/3}) $.
\begin{que}
What is the asymptotic behavior of $ f_{s,\lfloor \lambda s \rfloor}(n)$?
\end{que}

Recall that Erd\H{o}s~\cite{ER} asked if for fixed $s+2\le t$, $\lim_{n\to\infty} \frac{f_{s+1,t}(n)}{f_{s,t}(n)} = \infty$. We ask a similar question, that if answered in the affirmative would imply an affirmative answer to the question of Erd\H{o}s.
\begin{que}
For all $t>s\ge 3$, is $\lim_{n\to\infty} \frac{f_{s+1,t+1}(n)}{f_{s,t}(n)} = \infty$?
\end{que}

redpajama_set_name: RedPajamaArXiv


text:
Elizabeth Hendrickson Joins Y&R!
Soaps.com is thrilled to report that Elizabeth Hendrickson, best known for her role as All My Children's Maggie Stone is headed to Genoa City! The Young and the Restless has snagged Elizabeth to play a figure from the fashion/modeling world that will become an acquainted with Lily (Christel Khalil).
Elizabeth Hendrickson's popular character Maggie at AMC was involved in many front burner stories, the last of which saw her paired with her former best friend Bianca (Eden Riegel) as a love match! The character has not been seen on that show since February of 2007.
Elizabeth also brags a long list of acting credits as a guest star on popular television dramas such as "Cold Case", "CSI: Miami", and "Criminal Minds". She has also been featured in commercials for Special K, Papa Johns, and Marshalls. The beautiful actress is also intelligent; she studied at Fordham University, as well as at Stanford with a major in Musical Theater. Elizabeth Hendrickson hails from Northport, New York.
Soaps.com welcomes Elizabeth back to daytime and to The Young and the Restless. We can't wait to see what her character is all about! Watch this space, and Comings and Goings for more details as they become available!
Pat Benetar to Perform on Y&R!
Former Y&R Actress Eva Longoria in...


This is not an episode of Mabel. Find out more about our mysterious season-long competition, which will be wrapping up with the final episode of season three on Monday.
Après moi, le feu. Featuring: a garden, a leash, a mirror image. Music: Anxiety Remains by Ars Sonor, Get Burn by Slow Blood, and M, Volume II by (morse). Written and performed by Mabel Martin, produced by Becca De La Rosa.