Let and consist of sets of size at most , i.e. , it holds that .
Show that if is an antichain then .
(Due to Frankl 1986)
Let be a -uniform set family satisfying that for any .
Show that .
Hint: Use Sperner's theorem. Consider how to construct the antichain.
Let and be such a -uniform set family that for all .
How large can be at most?
Hint: Use Erdos-Ko-Rado theorem. Consider how to construct an intersecting family out of .
An -player tournament (竞赛图) is said to be transitive, if there exists a permutation of such that for every .
Show that for any , there exists an such that every tournament of players contains a transitive sub-tournament of players.
Hint: Use Ramsey Theorem.