FormalConjectures/ErdosProblems/862.lean

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L4DEL
Lean4 Domain Explanation Language

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 862

References:

Imports (1)
  1. def Erdos862.numMaximalSidonSets $A_1(N)$, the number of maximal Sidon subsets of $\{1, \dots, N\}$ line 40 source

    Source documentation

    $A_1(N)$, the number of maximal Sidon subsets of $\{1, \dots, N\}$.

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

    Statement uses: Nat

    Proof uses: Set.ncard, Finset, Set.ofPred, And, LE.le, Finset.Icc, OfNat.ofNat, Set.IsMaximalSidonSetIn

    8 instances, coercions and auxiliary constants left out.

    noncomputable def numMaximalSidonSets (N : ℕ) : ℕ :=
…
  2. theorem Erdos862.erdos_862.parts.i Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$ sorry line 54 source

    Source documentation

    Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$. Is it true that $$A_1(N) < 2^{o(N^{1/2})}?$$

    A problem of Cameron and Erdős. This is resolved as a consequence of results of Saxton and Thomason [SaTh15] - they prove that the number of Sidon sets in $\{1,\ldots,N\}$ is at least $2^{(1.16+o(1))N^{1/2}}$. Since each Sidon set is contained in a maximal Sidon set, and each maximal Sidon set contains at most $2^{(1+o(1))N^{1/2}}$ Sidon sets, it follows that $$A_1(N) \geq 2^{(0.16+o(1))N^{1/2}}.$$

    depends on sorry what this means

    Statement uses: Iff, False, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Real.logb, OfNat.ofNat, Erdos862.numMaximalSidonSets, HPow.hPow, HDiv.hDiv

    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.

    theorem erdos_862.parts.i : answer(False) ↔
    (fun N : ℕ => Real.logb 2 (numMaximalSidonSets N : ℝ)) =o[atTop]
      (fun N : ℕ => (N : ℝ) ^ (1 / 2 : ℝ)) := by
…

    Mentions: Erdos862.numMaximalSidonSets

  3. theorem Erdos862.erdos_862.parts.ii Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$ sorry line 71 source

    Source documentation

    Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$. Is it true that $$A_1(N) > 2^{N^c}$$ for some constant $c>0$?

    A problem of Cameron and Erdős. This is resolved as a consequence of results of Saxton and Thomason [SaTh15] - they prove that the number of Sidon sets in $\{1,\ldots,N\}$ is at least $2^{(1.16+o(1))N^{1/2}}$. Since each Sidon set is contained in a maximal Sidon set, and each maximal Sidon set contains at most $2^{(1+o(1))N^{1/2}}$ Sidon sets, it follows that $$A_1(N) \geq 2^{(0.16+o(1))N^{1/2}}.$$

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, HPow.hPow, Erdos862.numMaximalSidonSets, Filter.atTop

    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_862.parts.ii : answer(True) ↔
    ∃ c : ℝ, 0 < c ∧ ∀ᶠ N : ℕ in atTop,
      (2 : ℝ) ^ ((N : ℝ) ^ c) < (numMaximalSidonSets N : ℝ) := by
…

    Mentions: c, Erdos862.numMaximalSidonSets

  4. theorem Erdos862.erdos_862.variants.lower_bound This is resolved as a consequence of results of Saxton and Thomason [SaTh15] - they prove that the number of Sidon sets in $\{1,\ldots,N\}$ is at least $2^{(1.16+o(1))N^{1/2}}$ sorry line 84 source

    Source documentation

    This is resolved as a consequence of results of Saxton and Thomason [SaTh15] - they prove that the number of Sidon sets in $\{1,\ldots,N\}$ is at least $2^{(1.16+o(1))N^{1/2}}$. Since each Sidon set is contained in a maximal Sidon set, and each maximal Sidon set contains at most $2^{(1+o(1))N^{1/2}}$ Sidon sets, it follows that $$A_1(N) \geq 2^{(0.16+o(1))N^{1/2}}.$$

    depends on sorry what this means

    Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HPow.hPow, HMul.hMul, HSub.hSub, OfScientific.ofScientific, Bool.true, HDiv.hDiv, Erdos862.numMaximalSidonSets, Filter.atTop

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

    48 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_862.variants.lower_bound (ε : ℝ) (hε : 0 < ε) :
    ∀ᶠ N : ℕ in atTop,
      (2 : ℝ) ^ (((0.16 : ℝ) - ε) * (N : ℝ) ^ (1 / 2 : ℝ)) ≤ (numMaximalSidonSets N : ℝ) := by
…

    Mentions: Erdos862.numMaximalSidonSets