Ramsey numbers of grid graphs

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 $G_{2\times 2}$.