An application of the Ramsey theorem to ordered r-tournaments
An r-tournament consists of a linearly ordered n-set V and a linear ordering of each of its r-element subsets. The r-tournament is transitive (anti-transitive) if the order of each of its r-element subsets agrees with the ordering of V (agrees with the reverse of the ordering of V). Given an r-tournament with underlying set V and a subset U of V, the r- tournament on U, which assigns the induced order to U and assigns to any r-subset the order it was given as a subset of V, is called an r- subtournament. The main result of this paper is: Theorem. For every two positive integers r and n satisfying \(r\leq n\), there exists a positive integer f(r,n) with the following property: Every ordered r-tournament on f(r,n) vertices either contains a transitive r-subtournament on n vertices or an anti-transitive r-tournament on n vertices.
scientific article; zbMATH DE number 3957161