On the grid Ramsey problem and related questions

TL;DR

This paper proves that the known exponential bound for the cube lemma in Ramsey theory cannot be improved to polynomial, establishing a superpolynomial lower bound and discussing related problems.

Contribution

It provides a superpolynomial lower bound for the cube lemma's bound, showing that the exponential bound cannot be replaced by a polynomial one.

Findings

The lower bound for the cube lemma is superpolynomial in r.

The exponential bound in Shelah's proof is essentially tight.

Discussion of related open problems in Ramsey theory.

Abstract

The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales--Jewett numbers are primitive recursive. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. In its simplest form, this lemma says that if we color the edges of the Cartesian product $K_n \times K_n$ in $r$ colors then, for $n$ sufficiently large, there is a rectangle with both pairs of opposite edges receiving the same color. Shelah's proof shows that $n = r^{\binom{r+1}{2}} + 1$ suffices. More than twenty years ago, Graham, Rothschild and Spencer asked whether this bound can be improved to a polynomial in $r$. We show that this is not possible by providing a superpolynomial lower bound in $r$. We also discuss a number of related problems.