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On tournaments with a prescribed property. (English) Zbl 0794.05032
Summary: 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 leat \(k\) 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 P. Erdős, G. and E. Szekeres, R. Graham and J. H. Spencer, and B. Bollobás. In this paper, we present a lower bound on \(f(n,k)\) for the case of Paley tournaments.

05C20 Directed graphs (digraphs), tournaments