On the asymptotic behavior of online Ramsey numbers for stars, paths and cycles

TL;DR

This paper investigates the asymptotic behavior of online Ramsey numbers for various graph pairs, establishing convergence results for paths, cycles, and stars as the size of the second graph grows.

Contribution

It proves that certain normalized online Ramsey numbers for paths, cycles, and stars converge to specific constants as the second graph's size increases.

Findings

Normalized online Ramsey numbers for paths and cycles converge to constants.

Normalized online Ramsey numbers for stars and paths/cycles also converge to constants.

The limits depend on the fixed star size and the type of second graph.

Abstract

The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. Each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy that guarantees a red copy of $G$ or a blue copy of $H$ in at most $t$ rounds. We show that for every fixed $k$, there are constants $\lambda_1$ and $\lambda_2$ such that $\tilde{r}(P_k,P_n)/n$ and $\tilde{r}(P_k,C_n)/n$ converge to $\lambda_1$, and $\tilde{r}(K_{1,k},P_n)/n$ and $\tilde{r}(K_{1,k},C_n)/n$ converge to $\lambda_2$.