A multidimensional Ramsey Theorem

TL;DR

The paper develops a multidimensional Ramsey theory framework for Cartesian products of graphs, proving that in any $r$-coloring of the edges of $\square^d K_N$ there exists a copy of $\square^d K_n$ monochromatic in every direction once $N \ge r^{r^{C_d r n^d}}$, i.e., doubly exponential in $n^d$. Central to the approach is the introduction of $d$-consistent structures and density-type arguments that first yield a structured subproduct and then embed a monochromatic $\square^d K_n$, yielding explicit bounds with constants depending only on $d$. The results extend to the multidimensional Erdős–Szekeres problem, establishing doubly exponential bounds for all dimensions, and connect to lex-monotone arrays, improving several prior bounds. Overall, the work narrows the growth gap between lower and upper bounds in high-dimensional Ramsey-type problems and provides a unified, scalable method for deriving dimension-dependent doubly exponential bounds.

Abstract

Ramsey theory is a central and active branch of combinatorics. Although Ramsey numbers for graphs have been extensively investigated since Ramsey's work in the 1930s, there is still an exponential gap between the best known lower and upper bounds. For $k$-uniform hypergraphs, the bounds are of tower-type, where the height grows with $k$. Here, we give a multidimensional generalisation of Ramsey's Theorem to Cartesian products of graphs, proving that a doubly exponential upper bound suffices in every dimension. More precisely, we prove that for every positive integers $r,n,d$, in any $r$-colouring of the edges of the Cartesian product $\square^{d} K_N$ of $d$ copies of $K_N$, there is a copy of $\square^{d} K_n$ such that the edges in each direction are monochromatic, provided that $N\geq 2^{2^{C_drn^{d}}}$. As an application of our approach we also obtain improvements on the multidimensional Erdős-Szekeres Theorem proved by Fishburn and Graham $30$ years ago. Their bound was recently improved by Bucić, Sudakov, and Tran, who gave an upper bound that is triply exponential in four or more dimensions. We improve upon their results showing that a doubly expoenential upper bounds holds any number of dimensions.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of Theorem \ref{['thm:mainRamsey']} for $d=2$
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:mainRamsey']}