832.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 832

References:

erdosproblems.com/832
[AkSh16] Akolzin, Ilia and Shabanov, Dmitry, Colorings of hypergraphs with large number of colors. Discrete Math. (2016), 3020--3031.
[Al85] Alon, Noga, Hypergraphs with high chromatic number. Graphs Combin. (1985), 387--389.
[ChPe20] Cherkashin, Danila and Petrov, Fedor, Regular behavior of the maximal hypergraph chromatic number. SIAM J. Discrete Math. (2020), 1326--1333.

Imports (1)

FormalConjecturesUtil

theorem Erdos832.erdos_832 Let $r\geq 3$ and $k$ be sufficiently large in terms of $r$ sorry line 47 source
Source documentation
Let $r\geq 3$ and $k$ be sufficiently large in terms of $r$. Is it true that every $r$-uniform hypergraph with chromatic number $k$ has at least $$\binom{(r-1)(k-1)+1}{r}$$ edges, with equality only for the complete graph on $(r-1)(k-1)+1$ vertices?
This was disproved by Alon [Al85]. The validity of this conjecture for $r=3$ remains open.

depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Filter.Eventually, Finset, Fin, Finset.IsUniform, Finset.HasHypergraphChromaticNumber, And, Nat.choose, HAdd.hAdd, HMul.hMul, HSub.hSub, Finset.card, Eq, Exists, Finset.powersetCard, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_832 :
    answer(False) ↔ ∀ r : ℕ, 3 ≤ r → ∀ᶠ k : ℕ in atTop,
    ∀ (n : ℕ) (H : Finset (Finset (Fin n))),
      H.IsUniform r → H.HasHypergraphChromaticNumber k →
      ((r - 1) * (k - 1) + 1).choose r ≤ H.card ∧
      (H.card = ((r - 1) * (k - 1) + 1).choose r →
        ∃ S : Finset (Fin n), S.card = (r - 1) * (k - 1) + 1 ∧ H = S.powersetCard r) := by
…
```