FormalConjectures.ErdosProblems.«532»

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import FormalConjecturesUtil

Erdős Problem 532

References:

erdosproblems.com/532

[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (1973), 117-138.

[Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (1975), 295-310.

[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (1977), 43-72.

[Hi74] Hindman, Neil, Finite sums from sequences within cells of a partition of $\mathbb{N}$. J. Combinatorial Theory Ser. A (1974), 1-11.

namespace Erdos532

If $\mathbb{N}$ is 2-coloured then is there some infinite set $A\subseteq \mathbb{N}$ such that all finite subset sums$$ \sum_{n\in S}n$$(as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?

Asked by Graham and Rothschild. Proved by Hindman [Hi74] (for any number of colours).

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos532.lean"]
theorem erdos_532 :
    answer(True) ↔ ∀ (c : ℕ → Fin 2),
      ∃ A : Set ℕ, A.Infinite ∧
        ∃ color : Fin 2,
          ∀ S : Finset ℕ, S.Nonempty → ↑S ⊆ A →
            c (∑ n ∈ S, n) = color := by⊢ True ↔ ∀ (c : ℕ → Fin 2), ∃ A, A.Infinite ∧ ∃ color, ∀ (S : Finset ℕ), S.Nonempty → ↑S ⊆ A → c (∑ n ∈ S, n) = color
  sorryAll goals completed! 🐙

end Erdos532