On-line Ramsey theory

J. A. Grytczuk, M. Hałuszczak, Henry Kierstead

Research output: Contribution to journalArticle

19 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1-10
Number of pages10
JournalElectronic Journal of Combinatorics
Volume11
Issue number1 R
StatePublished - Sep 9 2004

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Ramsey Theory
Paint
Graph in graph theory
Game
Outerplanar Graph
Planar graph
Target
Class

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

Cite this

Grytczuk, J. A., Hałuszczak, M., & Kierstead, H. (2004). On-line Ramsey theory. Electronic Journal of Combinatorics, 11(1 R), 1-10.

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

In: Electronic Journal of Combinatorics, Vol. 11, No. 1 R, 09.09.2004, p. 1-10.

Research output: Contribution to journalArticle

Grytczuk, JA, Hałuszczak, M & Kierstead, H 2004, 'On-line Ramsey theory', Electronic Journal of Combinatorics, vol. 11, no. 1 R, pp. 1-10.
Grytczuk JA, Hałuszczak M, Kierstead H. On-line Ramsey theory. Electronic Journal of Combinatorics. 2004 Sep 9;11(1 R):1-10.
Grytczuk, J. A. ; Hałuszczak, M. ; Kierstead, Henry. / On-line Ramsey theory. In: Electronic Journal of Combinatorics. 2004 ; Vol. 11, No. 1 R. pp. 1-10.
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