Hereditary properties of tournaments

Abstract

A collection of unlabelled tournaments P is called a hereditary property if it is closed under isomorphism and under taking induced sub-tournaments. The speed of P is the function n -> |P_n|, where P_n = {T \in P : |V(T)| = n}. In this paper, we prove that there is a jump in the possible speeds of a hereditary property of tournaments, from polynomial to exponential speed. Moreover, we determine the minimal exponential speed, |P_n| = c^(n + o(n)), where c = 1.47... is the largest real root of the polynomial x^3 = x^2 + 1, and the unique hereditary property with this speed.