Ramsey numbers of grid graphs

TL;DR

This work studies off-diagonal grid Ramsey numbers $gr(H,K_k)$ for grid subgraphs $H$ inside $G_{N\times N}$ and links them to hypergraph Ramsey numbers. It shows the square $G_{2\times2}$ is exceptional, while broad families such as simple cycles with length at least five and many grid patterns built via duplication have $gr(H,K_k)=k^{O(1)}$, with bridging and generalized subdivision as key tools. A central mechanism is row/column duplication, which preserves polynomial growth across a wide class of patterns, and a conditional theory based on the Multicolor Erdős–Hajnal conjecture extends these results to two-row grids. The work further translates grid Ramsey bounds into hypergraph Ramsey consequences, yielding polynomial bounds like $R(C_t^{(3)},S_k^{(3)})=k^{O_t(1)}$ for $t\ge5$, thereby illuminating a polynomial-exponential dichotomy for grid-based Ramsey phenomena and guiding future investigations into grid-structure Ramsey theory.

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, Verstraëte, 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 $2^{Ω(\log ^2 k)} \le \operatorname{gr}(G_{2\times 2}, K_k) \le 2^{O(k^{2/3}\log k)}$. 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 Erdős-Hajnal conjecture, $\operatorname{gr}(H,K_k) = k^{O_H(1)}$ for any $H$ with two rows that does not contain $G_{2\times 2}$.

Figures (6)

• Figure 1: The alternating cycle $AC_{8}$ of length $8$.
• Figure 2: The tight cycle $C_8^{(3)}$ on 8 vertices.
• Figure 3: Dotted edges denote auxiliary edges.
• Figure 4: $AS_4$
• Figure :

Theorems & Definitions (43)

• Definition 1.1: Simple grid subgraph
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Conjecture 1.5: Multicolor Erdős-Hajnal conjecture (see Fox, Grinspuch, Pach FOX201575)
• Theorem 1.6
• Conjecture 1.7
• Definition 2.1: Embedding
• Definition 2.2: Vertical/horizontal degrees and neighborhoods
• Proposition 2.3: Existence of grid Ramsey numbers