Transitive Sets in Euclidean Ramsey Theory

TL;DR

This paper introduces a new conjecture in Euclidean Ramsey theory proposing that Ramsey sets are exactly the transitive sets and their subsets, supported by initial proofs and distinctions from previous spherical set conjectures.

Contribution

It proposes the first rival conjecture to the classical spherical set conjecture, linking Ramsey sets to transitive symmetry and providing initial evidence and related conjectures.

Findings

Proposed a new conjecture relating Ramsey sets to transitive sets.

Proved the first non-trivial cases of the conjecture.

Demonstrated that not all spherical sets embed in transitive sets.

Abstract

A finite set $X$ in some Euclidean space $R^n$ is called Ramsey if for any $k$ there is a $d$ such that whenever $R^d$ is $k$-coloured it contains a monochromatic set congruent to $X$. This notion was introduced by Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory. In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever `rival' conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively---in other words, if all points of the set look the same---our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement. Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest.