### Abstract

We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum,number of rounds (assuming both players play perfectly) is the on-line Ramsey number r̃(H) of the graph H. We determine exact values of r̃(H) for a few short paths and obtain a general upper bound r̃(P_{n}) ≤ 4n - 7. We also study asymmetric version of this parameter when one of the target graphs is a star S_{n} with n edges. We prove that r̃(S_{n}, H) ≤ n · e(H) when H is any tree, cycle or clique.

Original language | English (US) |
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Pages (from-to) | 63-74 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

Volume | 10 |

Issue number | 3 |

State | Published - Oct 13 2008 |

### Keywords

- Online Ramsey games
- Size Ramsey number

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics and Theoretical Computer Science*,

*10*(3), 63-74.