FormalConjectures/ErdosProblems/836.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 836

References:

Imports (1)
  1. theorem Erdos836.erdos_836.parts.i Let $r\geq 2$ and let $G$ be an intersecting $r$-uniform hypergraph with chromatic number $3$ sorry line 44 source

    Source documentation

    Let $r\geq 2$ and let $G$ be an intersecting $r$-uniform hypergraph with chromatic number $3$. Must $G$ contain $O(r^2)$ many vertices? Vertices are counted in the union of the edges, so isolated vertices are excluded.

    Alon has provided a counterexample to the first question: this hypergraph is intersecting, its chromatic number is $3$, and it has $\asymp 4^r/\sqrt{r}$ many vertices.

    depends on sorry what this means

    Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Finset, Fin, Finset.IsUniform, Finset.HasHypergraphChromaticNumber, Set.Intersecting, Finset.card, Finset.biUnion, id, HMul.hMul, HPow.hPow

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

    40 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_836.parts.i :
    answer(False) ↔ ∃ C : ℝ, 0 < C ∧ ∀ r : ℕ, 2 ≤ r →
    ∀ (n : ℕ) (H : Finset (Finset (Fin n))),
      H.IsUniform r → H.HasHypergraphChromaticNumber 3 → (H : Set (Finset (Fin n))).Intersecting →
        ((H.biUnion id).card : ℝ) ≤ C * (r : ℝ) ^ 2 := by
…
  2. theorem Erdos836.erdos_836.parts.ii In an intersecting $r$-uniform hypergraph with chromatic number $3$, must there be two distinct edges which meet in $\gg r$ many vertices? sorry line 56 source

    Source documentation

    In an intersecting $r$-uniform hypergraph with chromatic number $3$, must there be two distinct edges which meet in $\gg r$ many vertices?

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Finset, Fin, Finset.IsUniform, Finset.HasHypergraphChromaticNumber, Set.Intersecting, Membership.mem, Ne, HMul.hMul, Finset.card, Inter.inter

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

    36 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_836.parts.ii :
    answer(sorry) ↔ ∃ c : ℝ, 0 < c ∧ ∀ r : ℕ, 2 ≤ r →
    ∀ (n : ℕ) (H : Finset (Finset (Fin n))),
      H.IsUniform r → H.HasHypergraphChromaticNumber 3 → (H : Set (Finset (Fin n))).Intersecting →
        ∃ e ∈ H, ∃ f ∈ H, e ≠ f ∧ c * r ≤ ((e ∩ f).card : ℝ) := by
…

    Mentions: c