Avoider–Enforcer games

Let p and q be positive integers and let H be any hypergraph. In a (p,q,H) Avoider–Enforcer game two players, called Avoider and Enforcer, take turns selecting previously unclaimed vertices of H. Avoider selects p vertices per move and Enforcer selects q vertices per move. Avoider loses if he claims all the vertices of some hyperedge of H; otherwise Enforcer loses. We prove a sufficient condition for Avoider to win the (p,q,H) game. We then use this condition to show that Enforcer can win the (1,q) perfect matching game on K2n for every q⩽cn/logn for an appropriate constant c, and the (1,q) Hamilton cycle game on Kn for every q⩽cnloglogloglogn/lognlogloglogn for an appropriate constant c. We also determine exactly those values of q for which Enforcer can win the (1,q) connectivity game on Kn. This result is quite surprising as it substantially differs from its Maker–Breaker analog. Our method extends easily to improve a result of Lu [X. Lu, A note on biased and non-biased games, Discrete Appl. Math. 60 (1995) 285–291], regarding forcing an opponent to pack many pairwise edge disjoint spanning trees in his graph.