Abstract

A set X is called Euclidean Ramsey if, for any k and sufficiently large m, any k-colouring of R^m contains a monochromatic congruent copy of X. This notion was introduced by Erdos, 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. It was introduced in 2010 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. Similarly to the first question, the block sets conjecture remains very elusive. In this talk we discuss recent developments on the block sets conjecture and their implications to Euclidean Ramsey sets. Joint work with Imre Leader and Mark Walters