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Erdős problem 918

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A question of Erdős and Hajnal [ErHa68b], who proved that for every finite kk there is a graph with chromatic number ℵ1\aleph_1 and ℵk\aleph_k vertices where each subgraph on less than ℵk\aleph_k vertices has chromatic number ≤ℵ0\leq \aleph_0.

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9 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

918.lean

Retained formal statement6 of 9

In [ErHa69] the questions are stated with =ℵ0= \aleph_0 rather than ≤ℵ0\leq\aleph_0. This is a likely typo since it can be shown that no such graph exists in this case.

This is the second question with all subgraphs.

FormalConjectures/ErdosProblems/918.leanErdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.ii5 linesExact file
∀ (ω : Ordinal.{u}),  ¬∃ V G,      Cardinal.mk V = Cardinal.aleph (ω + 1) ∧        G.chromaticCardinal = Cardinal.aleph 1 ∧          ∀ (H : G.Subgraph), Cardinal.mk ↑H.verts = Cardinal.aleph ω → H.coe.chromaticCardinal = Cardinal.aleph0
TextbookStatement only, no proof