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 → |Pn|, where Pn = {T ∈ 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(1+o(1))n, where c ≃ 1:47 is the largest real root of the polynomial x3 = x2 + 1, and the unique hereditary property with this speed.