Acyclic sets in k-majority tournaments

Abstract

When Π is a set of k linear orders on a ground set X, and k is odd, the kmajority tournament generated by Π has vertex set X and has an edge from u to v if and only if a majority of the orders in Π rank u before v. Let fk(n) be the minimum, over all k-majority tournaments with n vertices, of the maximum order of an induced transitive subtournament. We prove that f3(n) ≥ √n always and that f3(n) ≤ 2√n - 1 when n is a perfect square. We also prove that f5(n) ≥ n1/4. For general k, we prove that nck ≤ fk(n) ≤ ndk(n), where ck = 3-(k-1)/2.