Question 51
In a chess tournament there are $n$ players (where $n \gt 1$ is odd), and every two players play against each other exactly once. It is known that exactly $n$ games end in a tie. For any set $S$ of players including $A$ and $B$, we say that $A$ *admires* $B$ in $S$ if
(a) $A$ does not beat $B$; or
(b) there exists a sequence of other distinct players $C_1, C_2, \dots, C_k$ in $S$ such that $A$ does not beat $C_1$; $C_k$ does not beat $B$; and $C_i$ does not beat $C_{i+1}$ for $1 \le i \le k-1$.
A set of four players is said to be *harmonic* if each of the four players admires everyone else in the set. Find (in terms of $n$) the greatest possible number of harmonic sets.
(a) $A$ does not beat $B$; or
(b) there exists a sequence of other distinct players $C_1, C_2, \dots, C_k$ in $S$ such that $A$ does not beat $C_1$; $C_k$ does not beat $B$; and $C_i$ does not beat $C_{i+1}$ for $1 \le i \le k-1$.
A set of four players is said to be *harmonic* if each of the four players admires everyone else in the set. Find (in terms of $n$) the greatest possible number of harmonic sets.