Monochromatic boxes in colored grids

TL;DR

This paper investigates conditions under which high-dimensional, multi-colored grids necessarily contain monochromatic boxes, providing asymptotically tight criteria and analyzing minimal configurations.

Contribution

It establishes asymptotically tight conditions for when a grid must contain a monochromatic box in a c-coloring, advancing understanding of hypergraph Ramsey problems.

Findings

Derived asymptotically tight conditions for c-guaranteed grids.

Characterized minimally c-guaranteed grids.

Connected the problem to hypergraph Ramsey theory and van der Waerden numbers.

Abstract

A $d$-dimensional grid is a set of the form $R = [a_1] \times ... \times [a_d]$. A $d$-dimensional box is a set of the form $\{b_1,c_1\} \times ... \times \{b_d,c_d\}$. When a grid is $c$-colored, must it admit a monochromatic box? If so, we say that $R$ is $c$-guaranteed. This question is a relaxation of one attack on bounding the van der Waerden numbers, and also arises as a natural hypergraph Ramsey problem (viz. the Ramsey numbers of hyperoctahedra). We give conditions on the $a_i$ for $R$ to be $c$-guaranteed that are asymptotically tight, and analyze the set of minimally $c$-guaranteed grids.