Lower bounds for online size Ramsey numbers for paths
Natalia Adamska, Grzegorz Adamski
TL;DR
The paper addresses online size Ramsey numbers for paths by developing a potential-function framework and Painter strategies that tightly control the per-move growth of a global potential. It derives new lower bounds $ ilde{r}(P_7,G)\ge 8v(G)/5-v_1(G)$, $\tilde{r}(P_8,G)\ge 18v(G)/11-v_1(G)$, and $\tilde{r}(P_9,G)\ge 5v(G)/3-v_1(G)$, with the notable corollary $\tilde{r}(P_9,P_n)\ge 5n/3-2$ and the existence of the limit $\lim_{n\to\infty}\tilde{r}(P_k,P_n)/n$ for fixed $k$. The core technique is a carefully designed potential function $f(G,T)$ based on vertex-type costs, together with a structured, symmetric Painter strategy that ensures $\Delta f$ is bounded (e.g., $12$, $20$, or $44$ depending on the game), preventing rapid appearance of the target red/blue structures. By combining three specialized games ($P_7$, $P_8$, $P_9$), the authors prove the main lower bounds and, as a consequence, establish the existence of asymptotic limits for bipartite host graphs, enriching the understanding of online versus size Ramsey numbers for paths. The work also provides a general lemma regarding the potential framework that may be of independent use for related online Ramsey analyses.
Abstract
Given two graphs $H_1$ and $H_2$, an online Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round Builder selects an edge and Painter colors it red or blue. Builder is trying to force Painter to create a red copy of $H_1$ or a blue copy of $H_2$ as soon as possible, while Painter's goal is the opposite. The online (size) Ramsey number $\tilde{r}(H_1,H_2)$ is the smallest number of rounds in the game provided Builder and Painter play optimally. Let $v(G)$ be the number of vertices in the graph $G$ and $v_1(G)$ be the number of vertices of degree 1 in $G$. We prove that if $G$ has no isolated vertices, then $\tilde{r}(P_7,G)\ge 8v(G)/5-v_1(G)$, $\tilde{r}(P_8,G)\ge 18v(G)/11-v_1(G)$ and $\tilde{r}(P_9,G)\ge 5v(G)/3-v_1(G)$. In particular $\tilde{r}(P_9,P_n)\ge 5n/3-2,$ which with known upper bound implies $\lim_{n\to\infty} \tilde{r}(P_9,P_n)/n=5/3.$ We also show that for any fixed $k$, $\lim_{n\to\infty} \tilde{r}(P_k,P_n)/n$ exists.