On Tournaments with a Prescribed Property

A round robin tournament on \(q\) players in which draws are not permitted is said to have property \(P(n, k)\) if each player in any subset of \(n\) players is defeated by at least \(k\) other players. We consider the problem of determining the minimum value \(F(n, k)\) such that every tournament of order \(q \geq F(n, k)\) has property \(P(n, k)\). The case \(k = 1\) has been studied by Erdős, G. and E. Szekeres, Graham and Spencer, and Bollobás. In this paper we present a lower bound on \(F(n, k)\) for the case of Paley tournaments.