Online Ramsey Theory for Planar Graphs

An online Ramsey game (G,H) is a game between Builder and Painter, alternating in turns. During each turn, Builder draws an edge, and Painter colors it blue or red. Builder’s goal is to force Painter to create a monochromatic copy of G, while Painter’s goal is to prevent this. The only limitation for Builder is that after each of his moves, the resulting graph has to belong to the class of graphs H. It was conjectured by Grytczuk, Ha luszczak, and Kierstead (2004) that if H is the class of planar graphs, then Builder can force a monochromatic copy of a planar graph G if and only if G is outerplanar. Here we show that the “only if” part does not hold while the “if” part does.