A threshold for the Maker-Breaker clique game

Abstract

We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k-clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win this game is at n− 2 k+1 , for all k > 3, thus proving a conjecture from Ref. [Stojakovic and Szabó, Random Struct ´Algor 26 (2005), 204–223]. More precisely, we conclude that there exist constants c,C > 0 such that when p > Cn− 2k+1 the game is Maker’s win a.a.s., and when p < cn− 2 k+1 it is Breaker’s win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker’s win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker’s win in the triangle game played on the edge set of G(n, p).