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In this article, we explore an arithmetic analogue of a long-standing open problem in graph theory: what can be said about the number of monochromatic additive configurations in 2-colourings of finite abelian groups? We are able to answer several instances of this question using techniques from additive combinatorics and quadratic Fourier analysis. However, the main purpose of this paper is to advertise this sphere of problems and to put forward a number of concrete questions and conjectures.

Ramsey multiplicity of linear patterns in certain finite abelian groups