Online Ramsey theory for a triangle on F‐free graphs

Abstract

Given a class of graphs and a fixed graph H, the online Ramsey game for H on is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in , and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of H at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of H forever, and we say Painter wins the game. We initiate the study of characterizing the graphs F such that for a given graph H, Painter wins the online Ramsey game for H on F‐free graphs. We characterize all graphs F such that Painter wins the online Ramsey game for C3 on the class of F‐free graphs, except when F is one particular graph. We also show that Painter wins the online Ramsey game forC3 on the class of K4‐minor‐free graphs, extending a result by Grytczuk, Hałuszczak, and Kierstead [Electron. J. Combin. 11 (2004), p. 60]. © 2019 Wiley Periodicals, Inc.