### 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) 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].

Original language | English (US) |
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Pages (from-to) | 423-427 |

Number of pages | 5 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 34 |

DOIs | |

State | Published - Aug 1 2009 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Keywords

- Positional games
- monotone property

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## Cite this

*Electronic Notes in Discrete Mathematics*,

*34*, 423-427. https://doi.org/10.1016/j.endm.2009.07.070