On-line Ramsey Theory

J. A. Grytczuk, M. Hałuszczak, H. A. Kierstead
<span title="2004-09-09">2004</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
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 ${\cal G}$. The main problem is to recognize the winner for a given pair $H,{\cal G}$. In particular, we prove that Builder has a winning
more &raquo; ... egy 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.
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