Imre Leader: Spherical sets versus transitive sets

The famous conjecture of Erdos, Graham, Montgomery, Rothschild, Spencer and Straus is that a set is Ramsey if and only if it embeds in a sphere. Here as usual `Ramsey' means that for every k there is an n such that whenever we k-colour real n-dimensional space there is a copy of the set that is monochromatic. We propose a `rival' conjecture, that a set is Ramsey if and only if it embeds into a (finite) transitive set. We will discuss the motivation for this conjecture, as well as how it relates to the original conjecture. (Joint work with Paul Russell and Mark Walters).