 
Summary: Playing to retain the advantage
Noga Alon
Dan Hefetz
Michael Krivelevich
March 11, 2009
Abstract
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)
MakerBreaker 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, Luczak and Ršodl [6].
1 Introduction
Let X be a finite set and let F 2X
be a family of subsets. In the (p : q)
MakerBreaker game (X, F), two players, called Maker and Breaker, take turns
