### 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) |
---|---|

Pages (from-to) | 49-64 |

Number of pages | 16 |

Journal | Combinatorica |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2009 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*29*(1), 49-64. https://doi.org/10.1007/s00493-009-2264-1

**Coloring number and on-line Ramsey theory for graphs and hypergraphs.** / Kierstead, Henry; Konjevod, Goran.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 29, no. 1, pp. 49-64. https://doi.org/10.1007/s00493-009-2264-1

}

TY - JOUR

T1 - Coloring number and on-line Ramsey theory for graphs and hypergraphs

AU - Kierstead, Henry

AU - Konjevod, Goran

PY - 2009/1

Y1 - 2009/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=65549158263&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65549158263&partnerID=8YFLogxK

U2 - 10.1007/s00493-009-2264-1

DO - 10.1007/s00493-009-2264-1

M3 - Article

AN - SCOPUS:65549158263

VL - 29

SP - 49

EP - 64

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -