Block Sizes in the Block Sets Conjecture

TL;DR

This paper investigates the block sets conjecture, demonstrating that block sizes cannot be bounded and establishing minimal block sizes for a specific template, advancing understanding of Euclidean Ramsey sets.

Contribution

It proves unboundedness of block sizes in the conjecture and identifies the minimal block size for the template 123, improving previous bounds from tower-type to linear.

Findings

Block sizes in the conjecture cannot be bounded, even over a 3-letter alphabet.

For the template 123, blocks can be of size 2 for any number of colours.

Previous bounds on block sizes were tower-type large.

Abstract

A set $X$ is called Euclidean Ramsey if, for any $k$ and sufficiently large $n$, every $k$-colouring of $\mathbb{R}^n$ contains a monochromatic congruent copy of $X$. This notion was introduced by Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus. They asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. It is not too difficult to show that if a set is not spherical then it is not Euclidean Ramsey either, but the converse is very much open despite extensive research over the years. On the other hand, the block sets conjecture is a purely combinatorial, Hales-Jewett type of statement, concerning `blocks in large products', introduced by Leader, Russell and Walters. If true, the block sets conjecture would imply that every transitive set (a set whose symmetry group acts transitively) is Euclidean Ramsey. As for the question above, the block sets conjecture remains very elusive, being known only in a few cases. In this paper we show that the sizes of the blocks in the block sets conjecture cannot be bounded, even for templates over the alphabet of size $3$. We also show that for the first non-trivial template, namely $123$, the blocks may be taken to be of size $2$ (for any number of colours). This is best possible; all previous bounds were `tower-type' large.