FormalConjectures/ErdosProblems/539.lean

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L4DEL
Lean4 Domain Explanation Language

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

10 declarations, 8 contain sorry, 0 declared as axioms

Erdős Problem 539

In this problem, a function $h : \mathbb{N} \to\mathbb{N}$ is defined maximally by a specified counting property.

The problem asks to estimate $h(n)$. This has been interpreted here as asking for $\Theta(h(n))$. The principal version includes answer(sorry) for an unknown function. On the other hand, the best known upper bound is $n^{2/3}$ and the best known lower bound is $\sqrt{n}$ so we also provide these candidates as variants. Moreover, it suffices to show $O(h(n))$ and $O(\sqrt{n})$ respectively for each, so further variants are provided for those.

In the source paper [Er73], Erdős also remarks that it should not be too difficult to determine $\lim_{n\to\infty}\log(h(n))/\log(n)$. This does not appear on the website, and it is not clear whether this remains open, but we include it here either way.

References:

Imports (1)
Imported by 1 file
  1. def Erdos539.IsCofactorLowerBound We say that $m$ is a cofactor lower bound for a given $n$ if, for every set $A$ of $n$ non-negative integers, there are at least $m$ cofactors $a / (a, b)$, where $a, b\in A$ line 53 source

    Source documentation

    We say that $m$ is a cofactor lower bound for a given $n$ if, for every set $A$ of $n$ non-negative integers, there are at least $m$ cofactors $a / (a, b)$, where $a, b\in A$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat

    Proof uses: Finset, Eq, Finset.card, LE.le, Finset.image, Prod, Erdos539.IsCofactorLowerBound.match_1, HDiv.hDiv, Nat.gcd, SProd.sprod

    5 instances, coercions and auxiliary constants left out.

    def IsCofactorLowerBound (n m : ℕ) : Prop := ∀ A : Finset ℕ, #A = n →
…
  2. def Erdos539.cofactorThreshold The cofactor threshold $h(n)$, for every positive $n$, is the largest cofactor lower bound for $n$ line 58 source

    Source documentation

    The cofactor threshold $h(n)$, for every positive $n$, is the largest cofactor lower bound for $n$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat

    Proof uses: SupSet.sSup, Set.ofPred, Erdos539.IsCofactorLowerBound

    1 instances, coercions and auxiliary constants left out.

    noncomputable def cofactorThreshold (n : ℕ) : ℕ :=
…
  3. theorem Erdos539.erdos_539 Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set$$\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$$has size at least $h(n)$ sorry line 64 source

    Source documentation

    Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set$$\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$$has size at least $h(n)$. Estimate $h(n)$.

    depends on sorry what this means

    Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos539.cofactorThreshold, sorryAx, Bool.false

    Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    9 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_539 :
    (fun n ↦ (cofactorThreshold n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…

    Mentions: Erdos539.cofactorThreshold

  4. theorem Erdos539.erdos_539.variants.sq sorry line 74 source

    depends on sorry what this means

    Statement uses: Iff, False, Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos539.cofactorThreshold, Real.sqrt

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    9 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

  5. theorem Erdos539.erdos_539.variants.sq_isBigO Erdős and Szemerédi proved that$$n^{1/2} \ll h(n)$$ sorry line 81 source

    Source documentation

    Erdős and Szemerédi proved that$$n^{1/2} \ll h(n)$$.

    depends on sorry what this means

    Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Real.sqrt, Erdos539.cofactorThreshold

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    9 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_539.variants.sq_isBigO :
    (fun n : ℕ ↦ √n) =O[atTop] fun n ↦ (cofactorThreshold n : ℝ) := by
…

    Mentions: Erdos539.cofactorThreshold

  6. theorem Erdos539.erdos_539.variants.isBigO_sq sorry line 89 source

    depends on sorry what this means

    Statement uses: Iff, False, Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos539.cofactorThreshold, Real.sqrt

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    9 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

  7. theorem Erdos539.erdos_539.variants.sq_cube_root sorry line 101 source

    depends on sorry what this means

    Statement uses: Iff, False, Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos539.cofactorThreshold, HPow.hPow, HDiv.hDiv, OfNat.ofNat

    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.

  8. theorem Erdos539.erdos_539.variants.isBigO_sq_cube_root Granville and Roesler [GR99] showed that $$h(n)\ll n^{2/3}$$ sorry line 109 source

    Source documentation

    Granville and Roesler [GR99] showed that $$h(n)\ll n^{2/3}$$.

    depends on sorry what this means

    Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos539.cofactorThreshold, HPow.hPow, HDiv.hDiv, OfNat.ofNat

    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_539.variants.isBigO_sq_cube_root :
    (fun n ↦ (cofactorThreshold n : ℝ)) =O[atTop] fun n ↦ (n : ℝ) ^ ((2 : ℝ) / 3) := by
…

    Mentions: Erdos539.cofactorThreshold

  9. theorem Erdos539.erdos_539.variants.sq_cube_root_isBigO sorry line 117 source

    depends on sorry what this means

    Statement uses: Iff, False, Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Erdos539.cofactorThreshold

    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.

  10. theorem Erdos539.erdos_539.variants.limit sorry line 133 source

    depends on sorry what this means

    Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Real.log, Erdos539.cofactorThreshold, Filter.atTop, nhds, OfNat.ofNat

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    30 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.