6. On the planet, there are 100 mutually hostile countries. To maintain peace, they decide to form several alliances, with the requirement that each alliance includes no more than 50 countries, and any two countries must be in at least one alliance. How many alliances are needed at a minimum?
(a) What is the minimum number of alliances that can be formed to meet the requirements?
(b) If an additional restriction is added, that the union of any two alliances contains no more than 80 countries, how would you answer the above question?
6. 6 alliances. Each country should join no less than 3 alliances, so the number of alliances is at least 6.
(a) Only by dividing 100 countries into 4 groups, each with 25 countries, and then through combinations, we get $C{ }_{4}^{2}=6$ alliances.
(b) Only by dividing 100 countries into 10 groups $a_{1}, a_{2}, a_{8}, a_{4}, a_{8}, b_{1}, b_{2}, b_{8}, b_{4}$, $b_{B}$, each with 10 countries, and forming the following alliances:
$$
\begin{array}{l}
a_{1} \cup b_{1} \cup b_{2} \cup b_{8} \cup a_{3}, \\
a_{2} \cup b_{2} \cup b_{3} \cup b_{4} \cup a_{4}, \\
a_{3} \cup b_{8} \cup b_{4} \cup b_{8} \cup a_{5}, \\
a_{1} \cup b_{4} \cup b_{8} \cup b_{1} \cup a_{1}, \\
a_{3} \cup b_{5} \cup b_{1} \cup b_{2} \cup a_{2}, \\
a_{1} \cup a_{2} \cup a_{9} \cup a_{4} \cup a_{8},
\end{array}
$$