Similarly, connect formula_128 with formula_126 if formula_130. Finally, connect formula_125 with formula_128 if formula_133. This construction is set up so that if formula_109 form a triangle, then we get elements formula_135 that all belong to formula_19. These numbers form an arithmetic progression in the listed order. The assumption on formula_19 then tells us this progression must be trivial: the elements listed above are all equal. But this condition is equivalent to the assertion that formula_109 is an arithmetic progression in formula_124. Consequently, every edge of formula_83 lies in exactly one triangle. The desired conclusion follows. formula_82  Szemerédi's theorem resolved the original conjecture and generalized Roth's theorem to arithmetic progressions of arbitrary length. Since then it has been extended in multiple fashions to create new and interesting results. Furstenberg and Katznelson used ergodic theory to prove a multidimensional version and Leibman and Bergelson extended it to polynomial progressions as well. Most recently Green and Tao proved the Green-Tao Theorem which says that the prime numbers contain arbitrarily long arithmetic progressions. Since the prime numbers are a subset of density 0, they introduced a "relative" Szemerédi theorem which applies to subsets with density 0 that satisfy certain pseudorandomness conditions. Later on Conlon, Fox, and Zhao strengthened this theorem by weakening the necessary pseudorandomness condition. There is another conjecture of Erdős asking whether any set formula_19 such that formula_143 diverges must contain arbitrarily long arithmetic progressions. There has also been work done on improving the bound in Roth's theorem. The bound from the original proof of Roth's theorem showed that  for some constant formula_145. Over the years this bound has been continually lowered by Szemerédi, Heath-Brown, Bourgain, and Sanders. The current best bound is due to Bloom who has shown that   There has also been work done on the other end, constructing the largest set with no three-term arithmetic progressions. The best construction has not been improved since 1946 when Behrend improved on the initial construction by Salem and Spencer and proved  so the gap between the two bounds is still quite large. A special case of Roth's Theorem on Arithmetic Progressions is the problem in the Finite Field setting. Consider the finite field formula_148, and let formula_149 be the size of the largest subset of formula_148 which contains no 3-term arithmetic progression.