Ramsey Multiplicity of Linear Patterns in Certain Finite Abelian Groups

Abstract. It is well known (and a result of Goodman) that a random 2-colouring of the edges of a large complete graph Kn contains asymptotically (in n) the minimum number of monochromatic triangles (K3s). Erdős conjectured that a random 2-colouring also minimises the number of monochromatic K4s, but his conjecture was disproved by Thomason in 1989. The question of determining for which small graphs Goodman’s result holds true remains wide open. In this article we explore an arithmetic analogue: what can be said about the number of monochromatic additive configurations in 2-colourings of finite abelian groups? While we are able to answer several instances of this question using techniques from additive combinatorics and quadratic Fourier analysis, the main purpose of this paper is to advertise this sphere of problems and to put forward a number of concrete conjectures. We also note that, perhaps surprisingly, some of our results in the arithmetic setting have implications for the original graph-theoretic problem.