Online size Ramsey numbers: Odd cycles vs connected graphs

TL;DR

This paper investigates the online size Ramsey numbers for odd cycles versus connected graphs, establishing bounds and exact values for specific cases, revealing new insights into the minimal rounds needed in these combinatorial games.

Contribution

It provides new bounds and exact results for the online size Ramsey numbers involving odd cycles and connected graphs, including a tight bound for the triangle versus path case.

Findings

Lower bound: ilde{r}( ext{odd cycles}, ext{connected graphs}) ext{ involves the golden ratio}

Upper bound: ilde{r}( ext{odd cycles}, ext{connected graphs}) ext{ expressed in terms of } n,m

Exact bound: ilde{r}(C_3, P_n) ext{ is between } 2.6n-3 ext{ and } 3n-4

Abstract

Given two graph families $\mathcal H_1$ and $\mathcal H_2$, a size Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder is trying to force Painter to create as soon as possible a red copy of a graph from $\mathcal H_1$ or a blue copy of a graph from $\mathcal H_2$. The online (size) Ramsey number $\tilde{r}(\mathcal H_1,\mathcal H_2)$ is the smallest number of rounds in the game provided Builder and Painter play optimally. We prove that if $\mathcal H_1$ is the family of all odd cycles and $\mathcal H_2$ is the family of all connected graphs on $n$ vertices and $m$ edges, then $\tilde{r}(\mathcal H_1,\mathcal H_2)\ge \varphi n + m-2\varphi+1$, where $\varphi$ is the golden ratio, and for $n\ge 3$, $m\le (n-1)^2/4$ we have $\tilde{r}(\mathcal H_1,\mathcal H_2)\le n+2m+O(\sqrt{m-n+1})$. We also show that $\tilde{r}(C_3,P_n)\le 3n-4$ for $n\ge 3$. As a consequence we get $2.6n-3\le \tilde{r}(C_3,P_n)\le 3n-4$ for every $n\ge 3$.