FormalConjectures.ErdosProblems.«965»

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import FormalConjecturesUtil

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


namespace Erdos965

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.

@[category research solved, AMS 3 5]
theorem declaration uses 'sorry'erdos_965 :
    answer(False) ↔ ∀ f : ℝ → Fin 2, ∃ A : Set ℝ, ¬ A.Countable ∧
      ∀ᵉ (a ∈ A) (b ∈ A) (c ∈ A) (d ∈ A), a ≠ b → c ≠ d → f (a + b) = f (c + d) := by⊢ False ↔ ∀ (f : ℝ → Fin 2), ∃ A, ¬A.Countable ∧ ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A, a ≠ b → c ≠ d → f (a + b) = f (c + d)
  sorryAll goals completed! 🐙

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

@[category research solved, AMS 3 5]
theorem declaration uses 'sorry'erdos_965.variants.generalization : answer(False) ↔
     ∀ᵉ (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) := by⊢ False ↔
  ∀ k ≥ 2,
    ∀ (f : ℝ → Fin 2),
      ∃ A, ¬A.Countable ∧ ∀ (s t : Finset ℝ), ↑s ⊆ A → ↑t ⊆ A → s.card = k → t.card = k → f (s.sum id) = f (t.sum id)
  sorryAll goals completed! 🐙


end Erdos965