Monochromatic symmetric subsets of abelian groups

DefinitionMonochrome symmetric subsets (Kourovka 15.77)#✓fully proved

The statement, written additively, that for every infinite abelian group \(G\), every cardinal \(\alpha < |G|\), every natural number \(n\) and every colouring \(\chi \colon G \to \{0, \dots, n - 1\}\), there are a set \(A \subseteq G\) of cardinality \(\alpha\) on which \(\chi\) is constant and an element \(g \in G\) such that \(2g - a \in A\) for every \(a \in A\); that is, \(A\) is a monochrome set symmetric with respect to \(g\).

def monochromeSymmetricSubsetExists : Prop :=
  ∀ (G : Type) [AddCommGroup G], Infinite G → ∀ α : Cardinal, α < Cardinal.mk G →
    ∀ (n : ℕ) (χ : G → Fin n), ∃ A : Set G, Cardinal.mk A = α ∧ (∃ c, ∀ a ∈ A, χ a = c) ∧
      ∃ g : G, ∀ a ∈ A, g + g - a ∈ A

Used by 1

Used byMonochromatic symmetric subsets in finite colourings (Jayadevan)

TheoremMonochromatic symmetric subsets in finite colourings (Jayadevan)#●ready to prove

Let \(G\) be an infinite abelian group, written additively, let \(\alpha\) be a cardinal with \(\alpha < |G|\), and let \(n\) be a natural number. For every colouring \(\chi \colon G \to \{0, \dots, n - 1\}\) there are a set \(A \subseteq G\) of cardinality \(\alpha\) on which \(\chi\) is constant and an element \(g \in G\) such that \(2g - a \in A\) for every \(a \in A\).

This answers Kourovka Problem 15.77 (I. V. Protasov) affirmatively; the Kourovka Notebook (21st issue) marks it solved, "Yes, it is true" (AI solution confirmed by I. V. Protasov).

public theorem_wanted monochromeSymmetricSubsetExists_holds : monochromeSymmetricSubsetExists

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 15.77 (I. V. Protasov); resolution: A. Jayadevan, Monochromatic symmetric subsets in finite colourings of infinite abelian groups, preprint, Zenodo record 22909420 (2026), AI-generated solution confirmed by I. V. Protasov, main theorem (for G infinite abelian and α < |G|, every finite colouring has a monochromatic S with |S| = α and 2g - S = S for some g); the Kourovka Notebook (21st issue) marks 15.77 solved, https://zenodo.org/records/22909420.

Uses 1

Statement usesMonochrome symmetric subsets (Kourovka 15.77)

1 fully proved1 ready to prove

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