On the Chvátal-Erdös Triangle Game

Given a graph G and positive integers n and q, let G(G;n, q) be the game played on the edges of the complete graph Kn 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 G(K3;n, q), Maker has a winning strategy if q < √ 2n + 2 − 5/2, and if q ≥ 2√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 G(K3;n, q) whenever q ≥ (2 − 1/24)√n. Moreover, we provide additional evidence supporting the belief that this bound can be further improved to ( √ 2 + o(1)) √ n.