The asymptotic of off-diagonal online Ramsey numbers for paths

Abstract

We prove that for every $k\ge 10$, the online Ramsey number for paths $P_k$ and $P_n$ satisfies $\tilde{r}(P_k,P_n)\ge \frac{5}{3}n+\frac{k}{9}-4$, matching up to a linear term in $k$ the upper bound recently obtained by Bednarska-Bzdęga (2024). In particular, this implies ${lim}_{n\to \infty }\frac{\tilde{r}(P_k,P_n)}{n}=\frac{5}{3}$, whenever $10\le k=o\left(n\right)$, disproving a conjecture by Cyman et al. (2015).