Probabilistic intuition holds for a class of small subgraph games

Abstract

Consider the following two-player game on the edges of $K_n$, the complete graph with $n$ vertices: Starting with an empty graph $G$ on the vertex set of $K_n$, in each round the first player chooses $b \in \mathbb{N}$ edges from $K_n$ which have not previously been chosen, and the second player immediately and irrevocably picks one of these edges and adds it to $G$. We show that for any graph $H$ with at least one edge, if $b < c n^{1/m(H)}$, where $c = c(H) > 0$ only depends on $H$ and $m(H)$ is the usual density function, then the first player can ensure the resulting graph $G$ contains $\Omega(n^{v(H)} / b^{e(H)})$ copies of $H$. The bound on $b$ is the best possible apart from the constant $c$ and shows that the density of the resulting graph for which it is possible to enforce the appearance of $H$ coincides with a threshold for the appearance in the Erd\H{o}s-R\'enyi random graph. This resolves a conjecture by Bednarska-Bzd\c{e}ga, Hefetz, and Luczak and provides a prominent class of games for which probabilistic intuition accurately predicts the outcome. The strategy of the first player is deterministic with polynomial running time, with the degree depending on the size of $H$.