# Online Ramsey theory for a triangle on $F$-free graphs

## TL;DR

This paper studies the online Ramsey game for triangles on classes of graphs that exclude certain subgraphs, characterizing when Painter can avoid creating monochromatic triangles and extending known results to minor-free graphs.

## Contribution

It characterizes all graphs F for which Painter wins the online Ramsey game for triangles on F-free graphs, except one case, and extends results to K4-minor-free graphs.

## Key findings

- Characterization of graphs F where Painter wins for C3 on F-free graphs.
- Extension of results to K4-minor-free graphs.
- Identification of a unique exceptional graph F.

## Abstract

Given a class $\mathcal{C}$ of graphs and a fixed graph $H$, the online Ramsey game for $H$ on $\mathcal C$ 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 $\mathcal C$, 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 $C_3$ on the class of $F$-free graphs, except when $F$ is one particular graph. We also show that Painter wins the online Ramsey game for $C_3$ on the class of $K_4$-minor-free graphs, extending a result by Grytczuk, Ha{\l}uszczak, and Kierstead.