A note on multidimensional Ramsey numbers

TL;DR

This paper improves bounds on multidimensional Ramsey numbers, showing that large enough Cartesian products contain monochromatic subcubes, advancing understanding in high-dimensional combinatorics and Ramsey theory.

Contribution

It provides a significantly improved exponential bound for the existence of monochromatic subcubes in multidimensional Ramsey problems.

Findings

Established a new bound $N > 2^{2^{c n^{d-1}}}$ for monochromatic $K_n^d$

Improved previous bounds by Girão, Kronenberg, and Scott

Enhanced bounds for a multidimensional Erdős–Szekeres theorem

Abstract

Fix integers $d,r\ge 2$ and suppose that the edge set of the $d$-fold Cartesian product of the $N$-clique $K_N^d$ is $r$-colored. We show that there is a copy of $K_n^d$ whose edges in each direction are monochromatic provided $N > 2^{2^{c n^{d-1}}}$, where $c$ depends only on $r$ and $d$. This improves the previous best exponent of $n^d$ proved by Gir\~ao, Kronenberg, and Scott while also improving the best known bound due to them for a multidimensional Erd\H os-Szekeres monotone subsequence theorem introduced by Fishburn and Graham.