Erdős problem 593 · problems.science

Which finite triple systems occur in every triple system of uncountable chromatic number? The claimed characterization: exactly those that, after removing isolated vertices, are linear, have every hyperedge-node of their Levi graph meeting a bridge, and have every Berge cycle even.

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12 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

593.lean

Retained formal statement5 of 10

Conjunction of the two open implications gives the conjectured characterization: if both obligatory_implies_two_colorable and two_colorable_implies_obligatory hold, then the characterization conjectured in erdos_593 (IsObligatory F ↔ F.IsTwoColorable) follows by elementary Iff manipulation.

FormalConjectures/ErdosProblems/593.leanErdos593.erdos_593.variants.implications_combine3 linesExact file

1(∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F → F.IsTwoColorable) →2  (∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), F.IsTwoColorable → IsObligatory F) →3    ∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F ↔ F.IsTwoColorable

TestStatement only, no proof

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