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 7→ |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, |Pn| = 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. Work supported by OTKA grant T049398 and NSF grants DMS-0302804, DMS-0603769 and DMS 0600303, and UIUC Campus Research Board 06139 and 07048. Work supported by ITR grant CCR-0225610 and ARO grant W911NF-06-1-0076. Work done whilst at The University of Memphis, and supported by a Van Vleet Memorial Doctoral Fellowship. the electronic journal of combinatorics 14 (2007), #R60 1