Monochromatic symmetric subsets in finite colourings of infinite abelian groups

We prove the assertion of Kourovka Problem 15.77 for every finite number of colours. If G is an infinite abelian group and α < |G|, every finite colouring of G has a monochromatic set S of cardinality exactly α such that 2g − S = S for some g ∈ G. For infinite α the argument uses a free-set lemma and a finite-colour directed-path lemma at α⁺; finite α follows from a ternary-cube embedding and the Hales–Jewett theorem.