FormalConjectures.ErdosProblems.«46»

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import FormalConjecturesUtil

Erdős Problem 46

References:

erdosproblems.com/46

[Cr03] Croot, III, Ernest S., On a coloring conjecture about unit fractions. Ann. of Math. (2) (2003), 545-556.

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

namespace Erdos46

Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?

The answer is yes, as proved by Croot [Cr03] - indeed, there are infinitely many disjoint such monochromatic solutions.

@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos46.lean"]
theorem erdos_46 :
    answer(True) ↔
    -- For any finite colouring of the integers
    ∀ (𝓒 : ℕ → ℕ), (Set.range 𝓒).Finite →
      -- there are integers `2 ≤ n₁ < ⋯ < n_k`
      ∃ S : Finset ℕ, (∀ n ∈ S, 2 ≤ n) ∧
        -- whose reciprocals sum to `1`
        ∑ n ∈ S, (1 / n : ℚ) = 1 ∧
        -- and which all have the same colour
        (𝓒 '' (S : Set ℕ)).Subsingleton := by⊢ True ↔ ∀ (𝓒 : ℕ → ℕ), (Set.range 𝓒).Finite → ∃ S, (∀ n ∈ S, 2 ≤ n) ∧ ∑ n ∈ S, 1 / ↑n = 1 ∧ (𝓒 '' ↑S).Subsingleton
  sorryAll goals completed! 🐙

Croot [Cr03] proved more: there are infinitely many disjoint such monochromatic solutions.

@[category research solved, AMS 5 11]
theorem erdos_46.variants.infinitely_many_disjoint :
    answer(True) ↔
    ∀ (𝓒 : ℕ → ℕ), (Set.range 𝓒).Finite →
      ∃ S : ℕ → Finset ℕ, (∀ i j, i ≠ j → Disjoint (S i) (S j)) ∧
        ∀ i, (∀ n ∈ S i, 2 ≤ n) ∧ ∑ n ∈ S i, (1 / n : ℚ) = 1 ∧
          (𝓒 '' (S i : Set ℕ)).Subsingleton := by⊢ True ↔
  ∀ (𝓒 : ℕ → ℕ),
    (Set.range 𝓒).Finite →
      ∃ S,
        (∀ (i j : ℕ), i ≠ j → Disjoint (S i) (S j)) ∧
          ∀ (i : ℕ), (∀ n ∈ S i, 2 ≤ n) ∧ ∑ n ∈ S i, 1 / ↑n = 1 ∧ (𝓒 '' ↑(S i)).Subsingleton
  sorryAll goals completed! 🐙

In [ErGr80] they also ask for a monochromatic representation of any $\frac{a}{b}>0$.

@[category research solved, AMS 5 11]
theorem erdos_46.variants.positive_rat :
    answer(True) ↔
    ∀ (𝓒 : ℕ → ℕ), (Set.range 𝓒).Finite → ∀ q : ℚ, 0 < q →
      ∃ S : Finset ℕ, (∀ n ∈ S, 2 ≤ n) ∧ ∑ n ∈ S, (1 / n : ℚ) = q ∧
        (𝓒 '' (S : Set ℕ)).Subsingleton := by⊢ True ↔
  ∀ (𝓒 : ℕ → ℕ),
    (Set.range 𝓒).Finite → ∀ (q : ℚ), 0 < q → ∃ S, (∀ n ∈ S, 2 ≤ n) ∧ ∑ n ∈ S, 1 / ↑n = q ∧ (𝓒 '' ↑S).Subsingleton
  sorryAll goals completed! 🐙

end Erdos46