## Abstract

Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G _{0}=(V,E _{0}), where E _{0}=∅ and V is determined by Builder. On the ith round Builder constructs a new edge e _{i} (distinct from previous edges) and sets G _{i} =(V,E _{i} ), where E _{i} =E _{i-1}∪{e _{i} }. Painter responds by coloring e _{i} with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K _{s} ^{t} , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges. We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K _{s} ^{t} ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H _{0} = (V,∅) with an arbitrary finite number of vertices and no edges. Let H _{i-1}=(V _{i-1},E _{i-1}) be the hypergraph constructed in the first i - 1 rounds. On the i-th round Presenter plays by presenting a p-subset P _{i} ⊆V _{i-1} and Chooser responds by choosing an s-subset X _{i} ⊆P _{i} . The vertices in P _{i} - X _{i} are discarded and the edge X _{i} added to E _{i-1} to form E _{i} . Presenter wins the survival game if H _{i} contains a copy of K _{s} ^{t} for some i. We show that for positive integers p,s,t with -astr-temp≤p, Presenter has a winning strategy.

Original language | English (US) |
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Pages (from-to) | 49-64 |

Number of pages | 16 |

Journal | Combinatorica |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2009 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics