FormalConjectures/ErdosProblems/20.lean

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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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, 2 contain sorry, 0 declared as axioms

Erdős Problem 20

References:

Imports (1)
Imported by 1 file
  1. def Erdos20.f Let $f(n,k)$ be minimal such that every $F$ family of $n$-uniform sets with $|F| \ge f(n,k)$ contains a $k$-sunflower line 38 source

    Source documentation

    Let $f(n,k)$ be minimal such that every $F$ family of $n$-uniform sets with $|F| \ge f(n,k)$ contains a $k$-sunflower.

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

    Statement uses: Nat

    Proof uses: InfSet.sInf, Set.ofPred, Set, And, Membership.mem, Eq, Set.ncard, LE.le, Exists, IsSunflower

    4 instances, coercions and auxiliary constants left out.

    noncomputable def f (n k : ℕ) : ℕ :=
…
  2. theorem Erdos20.f_0_1 line 43 source

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

    Statement uses: Eq, Nat, Erdos20.f, OfNat.ofNat

    Proof uses: IsLeast.csInf_eq, Set.ofPred, Set, And, Membership.mem, Set.ncard, LE.le, Exists, IsSunflower, And.intro, lowerBounds, Set.Nonempty, Set.nonempty_of_ncard_ne_zero, Exists.intro, Singleton.singleton, Eq.mpr, id, congr, congrArg, Eq.trans and 20 more

    39 instances, coercions and auxiliary constants left out.

    Tactic trace: 10 steps

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

  3. theorem Erdos20.erdos_20 Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$? sorry line 54 source

    Source documentation

    Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$?

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Nat, GT.gt, OfNat.ofNat, LT.lt, Erdos20.f, HPow.hPow

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

    12 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_20 : answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ n k, n > 0 → f n k < (c k) ^ n := by
…

    Mentions: c, Erdos20.f

  4. theorem Erdos20.erdos_20.variants.erdos_rado_bound sorry line 66 source

    depends on sorry what this means

    Statement uses: Nat, GT.gt, OfNat.ofNat, LE.le, Erdos20.f, HAdd.hAdd, HMul.hMul, HPow.hPow, HSub.hSub, Nat.factorial

    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.