FormalConjectures/ErdosProblems/732.lean

GitHub source file

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 732

References:

Imports (1)
  1. theorem Erdos732.erdos_732.parts.i Call a sequence sorry line 43 source

    Source documentation

    Call a sequence $1< X_1\leq \cdots \leq X_m\leq n$ block-compatible if there is a pairwise balanced block design $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $\lvert A_i\rvert=X_i$ for $1\leq i\leq m$. (A pairwise block design means that every pair in $\{1,\ldots,n\}$ is contained in exactly one of the $A_i$.) Are there necessary and sufficient conditions for $(X_i)$ to be block-compatible? Is there some constant $c>0$ such that for all large $n$ there are $$\geq \exp(c n^{1/2}\log n)$$ many block-compatible sequences for $\{1,\ldots,n\}$?

    depends on sorry what this means

    Statement uses: Nat, List, sorryAx, Bool.false, Iff, Membership.mem, Finset, Hypergraph.blockSizeProfiles

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

    5 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_732.parts.i :
    let P : ℕ → List ℕ → Prop := answer(sorry)
…
  2. theorem Erdos732.erdos_732.parts.ii Is there some constant $c>0$ such that for all large $n$ there are at least $\exp(c n^{1/2}\log n)$ block-compatible sequences? sorry line 56 source

    Source documentation

    Is there some constant $c>0$ such that for all large $n$ there are at least $\exp(c n^{1/2}\log n)$ block-compatible sequences?

    Alon has proved there are at least $2^{(\frac{1}{2}+o(1))n^{1/2}\log n}$ many sequences which are block-compatible for $n$.

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Real.exp, HMul.hMul, Real.sqrt, Real.log, Finset.card, List, Hypergraph.blockSizeProfiles, Filter.atTop

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

    19 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_732.parts.ii :
    answer(True) ↔ ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
    Real.exp (c * Real.sqrt n * Real.log n) ≤ (Hypergraph.blockSizeProfiles n).card := by
…

    Mentions: c, Hypergraph.blockSizeProfiles