On Avoider-Enforcer Games

In the Avoider-Enforcer game on the complete graph Kn, the players (Avoider and Enforcer) each take an edge in turn. Enforcer wins the game when he can require Avoider’s graph to have a given property P . The important parameter is τE(P), the most number of rounds required for Enforcer to win if Avoider plays with an optimal strategy (τE(P) = ∞ if Avoider can finish the game without creating a graph with property P). In this paper, let F be an arbitrary family of graphs and P be the property that a member of F is a subgraph or is an induced subgraph. We determine the asymptotic value of τE(P) when F contains no bipartite graph and establish that τE(P) = o(n ) if F contains a bipartite graph. The proof uses the game of JumbleG and the Szemerédi Regularity Lemma.