On the Chvátal-Erdős Triangle Game

Abstract

Given a graph $G$ and positive integers $n$ and $q$, let ${\bf G}(G;n,q)$ be the game played on the edges of the complete graph $K_n$ in which the two players, Maker and Breaker, alternately claim $1$ and $q$ edges, respectively. Maker's goal is to occupy all edges in some copy of $G$; Breaker tries to prevent it. In their seminal paper on positional games, Chvátal and Erdős proved that in the game ${\bf G}(K_3;n,q)$, Maker has a winning strategy if $q < \sqrt{2n+2}-5/2$, and if $q \geq 2\sqrt{n}$, then Breaker has a winning strategy. In this note, we improve the latter of these bounds by describing a randomized strategy that allows Breaker to win the game ${\bf G}(K_3;n,q)$ whenever $q \geq (2-1/24)\sqrt{n}$. Moreover, we provide additional evidence supporting the belief that this bound can be further improved to $(\sqrt{2}+o(1))\sqrt{n}$.