Playing to retain the advantage

Let P be a monotone decreasing graph property, let G=(V,E) be a graph, and let q be a positive integer. In this paper, we study the (1:q) Maker-Breaker game, played on the edges of G, in which Maker's goal is to build a graph that does not satisfy the property P. It is clear that in order for Maker to have a chance of winning, G must not satisfy P. We prove that if G is far from satisfying P, that is, if one has to delete sufficiently many edges from G in order to obtain a graph that satisfies P, then Maker has a winning strategy for this game. We also consider a different notion of being far from satisfying some property, which is motivated by a problem of Duffus, Łuczak and Rödl [D. Duffus, T. Łuczak and V. Rödl, Biased positional games on hypergraphs, Studia Scientarum Matematicarum Hung. 34 (1998), 141–149].