Ramsey Functions for Symmetric Subsets in Compact Abelian Groups

Abstract

Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA ⁻¹ g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.