# David Conlon: Euclidean Ramsey Theory.

A finite set*X ⊆ ℝ*is said to be Ramsey if for any natural number

^{k}*r ≥ 2*there exists

*N*such that any

*r*-colouring of the points of

*ℝ*contains a monochromatic isometric copy of

^{N}*X*. The study of problems of this variety originates with work of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus in the 1970s. The aim of this course will be to survey, with full proofs, the known results about Ramsey sets, from the result of Erdős et al. that any Ramsey set must be spherical up to Kriz's remarkable result that any set

*X*with a transitive solvable group of isometries is Ramsey.