43.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 43

Reference: erdosproblems.com/43

Imports (1)

FormalConjecturesUtil

abbrev Erdos43.f Let $f(N)$ be the maximum possible size of a Sidon set in $\{1,\ldots,N\}$ line 35 source
Source documentation
Let $f(N)$ be the maximum possible size of a Sidon set in $\{1,\ldots,N\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.maxSidonSubsetCard, Finset.Icc, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev f (N : ℕ) : ℕ := Finset.maxSidonSubsetCard (Finset.Icc 1 N)
…
```
Mentions: Finset.maxSidonSubsetCard
theorem Erdos43.erdos_43.parts.i Documented declaration: i sorry line 46 source
Source documentation
If $A$ and $B$ are Sidon sets in $\{1,\ldots,N\}$ with $(A-A)\cap(B-B)=\{0\}$, is it true that $$\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1)?$$
The answer is no; the Erdős Problems page notes that this follows from the solution to Erdős Problem 42.

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, Filter.Eventually, Nat, Finset, LE.le, Finset.Icc, OfNat.ofNat, IsSidon, Eq, Inter.inter, HSub.hSub, Singleton.singleton, HAdd.hAdd, Nat.choose, Finset.card, Erdos43.f, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_43.parts.i : answer(False) ↔
    ∃ C : ℝ, ∀ᶠ N in Filter.atTop, ∀ (A B : Finset ℕ),
      A ⊆ Finset.Icc 1 N →
      B ⊆ Finset.Icc 1 N →
      IsSidon (A : Set ℕ) →
      IsSidon (B : Set ℕ) →
      (A - A) ∩ (B - B) = {0} →
      ((A.card.choose 2 + B.card.choose 2 : ℕ) : ℝ) ≤ ((f N).choose 2 : ℝ) + C := by
…
```
Mentions: IsSidon, Erdos43.f
theorem Erdos43.erdos_43.parts.ii Documented declaration: ii sorry line 66 source
Source documentation
If $A$ and $B$ are equal-sized Sidon sets in $\{1,\ldots,N\}$ with $(A-A)\cap(B-B)=\{0\}$, can the bound be improved to $$\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2} \leq (1-c+o(1))\binom{f(N)}{2}$$ for some constant $c>0$?
The answer is no; the Erdős Problems page records a negative answer due to Barreto.

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Asymptotics.IsLittleO, Filter.atTop, Filter.Eventually, Finset, LE.le, Finset.Icc, IsSidon, Eq, Finset.card, Inter.inter, HSub.hSub, Singleton.singleton and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
62 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_43.parts.ii : answer(False) ↔
    ∃ᵉ (c > 0), ∃ o : ℕ → ℝ, o =o[Filter.atTop] (1 : ℕ → ℝ) ∧
    ∀ᶠ N in Filter.atTop, ∀ (A B : Finset ℕ),
      A ⊆ Finset.Icc 1 N →
      B ⊆ Finset.Icc 1 N →
      IsSidon (A : Set ℕ) →
      IsSidon (B : Set ℕ) →
      A.card = B.card →
      (A - A) ∩ (B - B) = {0} →
      ((A.card.choose 2 + B.card.choose 2 : ℕ) : ℝ) ≤
        (1 - c + o N) * ((f N).choose 2 : ℝ) := by
…
```
Mentions: c, IsSidon, Erdos43.f