### On-line Ramsey Theory for Bounded Degree Graphs

#### Abstract

When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph $G$. In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class ${\cal H}$. Builder wins the game $(G,{\cal H})$ if a monochromatic copy of $G$ can be forced. The *on-line degree Ramsey number* $\mathring {R}_\Delta(G)$ is the least $k$ such that Builder wins $(G,{\cal H})$ when ${\mathcal H}$ is the class of graphs with maximum degree at most $k$. Our results include: 1) $\mathring {R}_\Delta(G)\!\le\!3$ if and only if $G$ is a linear forest or each component lies inside $K_{1,3}$. 2) $\mathring {R}_\Delta(G)\ge \Delta(G)+t-1$, where $t=\max_{uv\in E(G)}\min\{d(u),d(v)\}$. 3) $\mathring {R}_\Delta(G)\le d_1+d_2-1$ for a tree $G$, where $d_1$ and $d_2$ are two largest vertex degrees. 4) $4\le \mathring {R}_\Delta(C_n)\le 5$, with $\mathring {R}_\Delta(C_n)=4$ except for finitely many odd values of $n$. 5) $\mathring {R}_\Delta(G)\le6$ when $\Delta(G)\le 2$.

The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays "consistently".