Graph parameters, Ramsey theory and the speed of hereditary properties

Abstract

The speed of a hereditary property $P$ is the number $P_n$ of $n$-vertex labelled graphs in $P$. It is known that the rates of growth of $P_n$ constitute discrete layers and the speed jumps, in particular, from constant to polynomial, from polynomial to exponential and from exponential to factorial. One more jump occurs when the entropy $\lim_{n\to\infty}\frac{\log_2 P_n}{\binom{n}{2}}$ changes from 0 to a nonzero value. In the present paper, for each of these jumps we identify a graph parameter responsible for it, i.e. we show that a jump of the speed coincides with a jump of the respective parameter from finitude to infinity. In particular, we show that the speed of a hereditary property $P$ is sub-factorial if and only if the neighbourhood diversity of graphs in $P$ is bounded by a constant, and that the entropy of a hereditary property $P$ is 0 if and only if the VC-dimension of graphs in $P$ is bounded by a constant. All the result are obtained by Ramsey-type arguments.