Consider a tournament with $n$ contestants - that is, a complete graph directed graph $K_n$ where each edge is pointed one way or the other. We call a subset $\{a,b,c\}$ a "cyclic triplet" if each of the three wins one of the two games against the other two. It is not hard to find a maximum number of cyclic triplets.
We can argue by considering the triplets involving a particular contestant that the maximum number of "cyclic triplets" among such triplets occurs when he beats half of the remaining contestants. Hence the global maximum occurs when everyone beats half the remaining ones.
Now, define a subset $\{a,b,c,d\}$ to be a "cyclic quadruplet" if two contestants win two games against the other two, while the remaining two win one game. What is the maximum number of cyclic quadruplets?
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