Canonical Euclidean Ramsey theory

Speaker: Benedict Randall Shaw (Cambridge)
Series: Combinatorics Seminar

A set of points in Euclidean space (or 'configuration') C is said to be Ramsey if, for each r, there is some configuration S (possibly in higher-dimensional space) such that any r-colouring of S contains a set of points in the same colour congruent to C (a 'monochromatic' copy). The question of which configurations are Ramsey is of central importance to Euclidean Ramsey theory. Of course, if we do not bound the number of colours, then we can of course avoid this, by colouring every point of S differently (a 'rainbow' colouring of S). We therefore study a variant notion: we say C is canonically Ramsey if there is some fixed configuration S such that any colouring at all of S contains either a monochromatic or rainbow copy of C. We explain recent progress on the question of which configurations are canonically Ramsey.