834.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 834

References:

erdosproblems.com/834
[Li25] R. Li, On an Erdős-Lovász problem: $3$-critical $3$-graphs of minimum degree $7$. arXiv:2512.24850 (2025).

Imports (1)

FormalConjecturesUtil

theorem Erdos834.erdos_834.parts.i Documented declaration: i sorry line 44 source
Source documentation
Does there exist a $3$-critical $3$-uniform hypergraph in which every vertex has degree $\geq 7$, where criticality means that the transversal number is $3$ and deleting any edge reduces it to $2$?
In the first formulation, the transversal notion of criticality, Li [Li25] proves that a $3$-critical $3$-uniform hypergraph must have a vertex of degree $\leq 6$.

depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, Finset, Fin, And, Finset.IsUniform, OfNat.ofNat, Set.HasFiniteTransversal, Not, Membership.mem, Finset.erase, LE.le, Finset.hypergraphDegree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_834.parts.i :
    answer(False) ↔ ∃ (n : ℕ) (H : Finset (Finset (Fin n))),
    H.IsUniform 3 ∧ (H : Set (Finset (Fin n))).HasFiniteTransversal 3 ∧
      ¬ (H : Set (Finset (Fin n))).HasFiniteTransversal 2 ∧
      (∀ e ∈ H, (H.erase e : Set (Finset (Fin n))).HasFiniteTransversal 2) ∧
      ∀ v : Fin n, 7 ≤ H.hypergraphDegree v := by
…
```
theorem Erdos834.erdos_834.parts.ii Documented declaration: ii sorry line 60 source
Source documentation
Does there exist a $3$-critical $3$-uniform hypergraph with minimum degree at least $7$, where criticality means that deleting any edge or vertex reduces chromatic number $3$ to $2$?
In the second formulation, the chromatic notion of criticality, Li [Li25] provides an explicit $3$-critical $3$-uniform hypergraph on $9$ vertices with minimum degree $7$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Finset, Fin, And, Finset.IsUniform, OfNat.ofNat, Finset.HasHypergraphChromaticNumber, Membership.mem, Finset.HypergraphColorable, Finset.erase, Finset.filter, Not, LE.le, Finset.hypergraphDegree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_834.parts.ii :
    answer(True) ↔ ∃ (n : ℕ) (H : Finset (Finset (Fin n))),
    H.IsUniform 3 ∧ H.HasHypergraphChromaticNumber 3 ∧
      (∀ e ∈ H, (H.erase e).HypergraphColorable 2) ∧
      (∀ v : Fin n, (H.filter (fun e ↦ v ∉ e)).HypergraphColorable 2) ∧
      ∀ v : Fin n, 7 ≤ H.hypergraphDegree v := by
…
```