Hereditary properties of tournaments

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 ∈ 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.