Erdős Problem 965

For every 2-coloring of ℝ, is there an uncountable set $A ⊆ ℝ$ such that all sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour?

References:

• erdosproblems.com/965
• [Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.
• [HLS17] Hindman, Neil and Leader, Imre and Strauss, Dona, Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg. (2017), 275--287.
• [Ko16] Komjáth, Péter, A certain 2-coloring of the reals. Real Anal. Exchange (2016), 227--231.
• [SWCol] Sokoup Dániel and Weiss, William, Sums and Anti-Ramsey Colourings of ℝ. https://danieltsoukup.github.io/academic/finset_colouring.pdf

theorem Erdos965.erdos_965 : False ↔ ∀ (f : ℝ → Fin 2), ∃ (A : Set ℝ), ¬A.Countable ∧ ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A, a ≠ b → c ≠ d → f (a + b) = f (c + d)

Erdős asks in [Er75b] if for every 2-coloring of ℝ, there is an uncountable set $A ⊆ ℝ$ such that all sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour.

In [Ko16] Péter Komjáth constructed a counterexample. The same result was proven independently in [SWCol] by Sokoup and Weiss.

theorem Erdos965.erdos_965.generalization : False ↔ ∀ k ≥ 2, ∀ (f : ℝ → Fin 2), ∃ (A : Set ℝ), ¬A.Countable ∧ ∀ (s t : Finset ℝ), ↑s ⊆ A → ↑t ⊆ A → s.card = k → t.card = k → f (s.sum id) = f (t.sum id)

In fact, in both [Ko16] and [SWCol] a generalized example for $k$-sums is constructed.