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OP-8659Status unassessedLogic and foundations

Compatcness properties of graphs · Question 55

From ℵ2 to infinity · Historical workshop question

MSC 03 · Mathematical logic and foundations

Editorial mapping of AIMPL source category: Logic. http://aimpl.org/alephtwo/ Not an author-supplied MSC code.

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
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Problem statement

Let ϕ(n)\phi(n) be the statement ``for every graph of size ℵω+1\aleph_{\omega+1}, if every subgraph of size <ℵω+1<\aleph_{\omega+1} has chromatic number ≤ℵn\le\aleph_n, then the entire graph has chromatic number ℵn\aleph_n.''

Is ϕ(0)\phi(0) consistent?

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Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

It is known that ϕ(n)\phi(n) is consistent whenever 1≤n<ω1\le n<\omega. In fact, the sentence ∀n∈ω(n>0→ϕ(n))\forall n\in \omega(n>0\to \phi(n)) is consistent.

References
  1. [1]From ℵ2 to infinity link

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