Ramsey numbers of grid graphs

TL;DR

This paper investigates the Ramsey numbers of grid graphs, revealing that most have polynomial bounds except for the 2x2 grid, which exhibits exponential growth, and explores conditions under which these bounds hold.

Contribution

It demonstrates that the 2x2 grid graph is unique in having polynomial Ramsey bounds, extends results to other grid subgraphs, and links these bounds to the multicolor Erdős-Hajnal conjecture.

Findings

The 2x2 grid graph has polynomial Ramsey bounds.

Most grid subgraphs have polynomial bounds, except G_{2x2}.

Conditional on a conjecture, all two-row grid graphs without G_{2x2} have polynomial bounds.

Abstract

Let the grid graph $G_{M\times N}$ denote the Cartesian product $K_M \square K_N$. For a fixed subgraph $H$ of a grid, we study the off-diagonal Ramsey number $\operatorname{gr}(H, K_k)$, which is the smallest $N$ such that any red/blue edge coloring of $G_{N\times N}$ contains either a red copy of $H$ (a copy must preserve each edge's horizontal/vertical orientation), or a blue copy of $K_k$ contained inside a single row or column. Conlon, Fox, Mubayi, Suk, Verstra\"ete, and the first author recently showed that such grid Ramsey numbers are closely related to off-diagonal Ramsey numbers of bipartite $3$-uniform hypergraphs, and proved that $2^{\Omega(\log ^2 k)} \le \operatorname{gr}(G_{2\times 2}, K_k) \le 2^{O(k^{2/3}\log k)}$. We prove that the square $G_{2\times 2}$ is exceptional in this regard, by showing that $\operatorname{gr}(C,K_k) = k^{O_C(1)}$ for any cycle $C \ne G_{2\times 2}$. We also obtain that a larger class of grid subgraphs $H$ obtained via a recursive blowup procedure satisfies $\operatorname{gr}(H,K_k) = k^{O_H(1)}$. Finally, we show that conditional on the multicolor Erd\H{o}s-Hajnal conjecture, $\operatorname{gr}(H,K_k) = k^{O_H(1)}$ for any $H$ with two rows that does not contain $G_{2\times 2}$.