Monochromatic pairwise sums in the reals

DefinitionErdős Problem 965#✓fully proved

The statement that for every colouring of the real numbers with two colours there is a set \(A \subseteq \mathbb{R}\) of cardinality \(\aleph_1\) such that all sums \(a + b\) with \(a, b \in A\) and \(a \ne b\) have the same colour.

def alephOneMonochromaticPairSums : Prop :=
  ∀ c : ℝ → Fin 2, ∃ A : Set ℝ, Cardinal.mk A = Cardinal.aleph 1 ∧ ∃ i : Fin 2,
    ∀ a ∈ A, ∀ b ∈ A, a ≠ b → c (a + b) = i

Used by 1

Used byErdős Problem 965 is false

TheoremErdős Problem 965 is false#●ready to prove

It is not the case that for every colouring of the real numbers with two colours there is a set \(A \subseteq \mathbb{R}\) of cardinality \(\aleph_1\) such that all sums \(a + b\) with \(a, b \in A\) and \(a \ne b\) have the same colour.

Komjáth, and independently Soukup and Weiss, constructed in ZFC a \(2\)-colouring of \(\mathbb{R}\) such that for every uncountable \(A \subseteq \mathbb{R}\), in particular every \(A\) of cardinality \(\aleph_1\), the sums \(a + b\) of distinct \(a, b \in A\) take both colours.

public theorem_wanted alephOneMonochromaticPairSums_refuted : ¬ alephOneMonochromaticPairSums

Sources

• T. F. Bloom, Erdős Problems, Problem #965, https://www.erdosproblems.com/965 (status DISPROVED (LEAN)); resolution: P. Komjáth, A certain 2-coloring of the reals, Real Anal. Exchange (2016), 227-231, erdosproblems.com/965: Komjáth, and independently Soukup and Weiss, proved without the continuum hypothesis that some 2-colouring of ℝ makes the sums of k distinct elements of any uncountable A ⊆ ℝ non-monochromatic, for every k ≥ 1 (the case k = 2 disproves the question), https://www.erdosproblems.com/965.

Uses 1

Statement usesErdős Problem 965

1 fully proved1 ready to prove

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