46.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 3 contain sorry, 0 declared as axioms

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).

Imports (1)

FormalConjecturesUtil

theorem Erdos46.erdos_46 Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$? sorry line 43 source
Source documentation
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.

depends on sorry what this means
Statement uses: Iff, True, Nat, Set.Finite, Set.range, Exists, Finset, And, Membership.mem, LE.le, OfNat.ofNat, Eq, Rat, Finset.sum, HDiv.hDiv, Set.Subsingleton, Set.image
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
theorem Erdos46.erdos_46.variants.infinitely_many_disjoint Croot [Cr03] proved more: there are infinitely many disjoint such monochromatic solutions sorry line 59 source
Source documentation
Croot [Cr03] proved more: there are infinitely many disjoint such monochromatic solutions.

depends on sorry what this means
Statement uses: Iff, True, Nat, Set.Finite, Set.range, Exists, Finset, And, Ne, Disjoint, Membership.mem, LE.le, OfNat.ofNat, Eq, Rat, Finset.sum, HDiv.hDiv, Set.Subsingleton, Set.image
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
theorem Erdos46.erdos_46.variants.positive_rat In [ErGr80] they also ask for a monochromatic representation of any $\frac{a}{b}>0$ sorry line 71 source
Source documentation
In [ErGr80] they also ask for a monochromatic representation of any $\frac{a}{b}>0$.

depends on sorry what this means
Statement uses: Iff, True, Nat, Set.Finite, Set.range, Rat, LT.lt, OfNat.ofNat, Exists, Finset, And, Membership.mem, LE.le, Eq, Finset.sum, HDiv.hDiv, Set.Subsingleton, Set.image
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```