Online Ramsey numbers: Long versus short cycles

TL;DR

This paper investigates the online Ramsey numbers for cycles, showing different bounds depending on whether the cycle length is even or odd, with implications for understanding online Ramsey game dynamics.

Contribution

It provides new bounds for the online Ramsey numbers involving cycles, distinguishing between even and odd fixed cycle lengths.

Findings

For even k,  r̃(C_k,C_n) = 2n + O(k)

For odd k,  r̃(C_k,C_n)   3n + o(n)

Bounds depend on the parity of the fixed cycle length k.

Abstract

Online Ramsey game is played between Builder and Painter on an infinite board $K_{\mathbb N}$. In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs $H_1$ and $H_2$. Builder aims to create either a red copy of $H_1$ or a blue copy of $H_2$ in $K_{\mathbb N}$ as soon as possible, and Painter tries to prevent it. The online Ramsey number $\tilde{r}(H_1,H_2)$ is the minimum number of rounds such that the Builder wins. We study $\tilde{r}(C_k,C_n)$ where $k$ is fixed and $n$ is large. We show that $\tilde{r}(C_k,C_n)=2n+\mathcal O(k)$ for an absolute constant $c$ if $k$ is even, while $\tilde{r}(C_k,C_n)\le 3n+o(n)$ if $k$ is odd.