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 statement6 of 10

No hyperedges implies chromatic cardinal ≤ 1: A 3-uniform hypergraph with no edges can be properly colored with a single color, so its chromatic cardinal is at most 1. In particular, $\chi(H) > \aleph_0$ implies H has at least one hyperedge.

FormalConjectures/ErdosProblems/593.leanErdos593.erdos_593.variants.nonempty_edges_if_large_chromatic1 lineExact file

1∀ {V : Type} (H : ThreeUniformHypergraph V), Cardinal.aleph0 < H.chromaticCardinal → H.edges.Nonempty

TextbookStatement only, no proof

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