### Abstract

The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph H, keeping the constructed graph in a prescribed class G. The main problem is to recognize the winner for a given pair H, G. In particular, we prove that Builder has a winning strategy for any k-colorable graph H in the game played on k-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for 3-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.

Original language | English (US) |
---|---|

Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Electronic Journal of Combinatorics |

Volume | 11 |

Issue number | 1 R |

State | Published - Sep 9 2004 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Electronic Journal of Combinatorics*,

*11*(1 R), 1-10.

**On-line Ramsey theory.** / Grytczuk, J. A.; Hałuszczak, M.; Kierstead, Henry.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 11, no. 1 R, pp. 1-10.

}

TY - JOUR

T1 - On-line Ramsey theory

AU - Grytczuk, J. A.

AU - Hałuszczak, M.

AU - Kierstead, Henry

PY - 2004/9/9

Y1 - 2004/9/9

N2 - The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph H, keeping the constructed graph in a prescribed class G. The main problem is to recognize the winner for a given pair H, G. In particular, we prove that Builder has a winning strategy for any k-colorable graph H in the game played on k-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for 3-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.

AB - The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph H, keeping the constructed graph in a prescribed class G. The main problem is to recognize the winner for a given pair H, G. In particular, we prove that Builder has a winning strategy for any k-colorable graph H in the game played on k-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for 3-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.

UR - http://www.scopus.com/inward/record.url?scp=5344225311&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=5344225311&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:5344225311

VL - 11

SP - 1

EP - 10

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1 R

ER -