On Sets of Consistent Arcs in a Tournament

Abstract

A (round-robin) tournament T n consists n of nodes u 1 , u 2 , …, u n such that each pair of distinct nodes u i and u j is joined by one of the (oriented) arcs or The arcs in some set S are said to be consistent if it is possible to relabel the nodes of the tournament in such a way that if the arc is in S then i>j. (This is easily seen to be equivalent to requiring that the tournament contains no oriented cycles composed entirely of arcs of S.) Sets of consistent arcs are of interest, for example, when the tournament represents the outcome of a paired-comparison experiment [1]. The object in this note is to obtain bounds for f(n), the greatest integer k such that every tournament T n contains a set of k consistent arcs.