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module
public import FormalConjecturesUtilErdős 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:
[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
@[expose] public sectionnamespace Erdos965Erdős 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,
formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos965.lean#L42"]
theorem 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) := ⊢ 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)
All goals completed! 🐙For every 2-coloring of ℝ, is there some $k ≥ 2$ and an uncountable set $A ⊆ ℝ$ such that all sums $a_1 + \cdots + a_k$ of $k$ distinct elements of $A$ have the same colour?
In fact, in both [Ko16] and [SWCol] a single 2-coloring of ℝ is constructed such that for every $k ≥ 2$ and every uncountable $A ⊆ ℝ$ the sums of $k$ distinct elements of $A$ are not monochromatic.
@[category research solved, AMS 3 5]
theorem erdos_965.variants.generalization : answer(False) ↔
∀ f : ℝ → Fin 2, ∃ k ≥ 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) := ⊢ False ↔
∀ (f : ℝ → Fin 2),
∃ k ≥ 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)
All goals completed! 🐙end Erdos965