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 K_{2 n} for every q ≤ c n / log n for an appropriate constant c, and the (1, q) Hamilton cycle game on K_n for every q ≤ c n log log log log n / log n log log log n for an appropriate constant c. We also determine exactly those values of q for which Enforcer can win the (1, q) connectivity game on K_n. 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.