Disproof of a conjecture of Erdös and moser on tournaments

Abstract

Erdös and Moser [1] displayed a tournament of order 7 with no transitive subtournament of order 4 and conjectured for each positive integer k existence of a tournament of order 2^k−1−1 with no transitive subtournament of order k. The conjecture is disproved for k=5. Further, every tournament of order 14 has a transitive subtournament of order 5. Inductively, the conjecture is false for all orders above 5. Existence and uniqueness of a tournament of order 13 having no transitive subtournament of order 5 are shown.