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OP-9389OpenLogic and foundations

Erdős #1175 · set theory / chromatic number

Let κ\kappa be an uncountable cardinal. Must there exist a cardinal λ\lambda such that every graph with chromatic number λ\lambda contains a triangle-free subgraph with chromatic number κ\kappa?

MSC 03 · Mathematical logic and foundations

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let κ\kappa be an uncountable cardinal. Must there exist a cardinal λ\lambda such that every graph with chromatic number λ\lambda contains a triangle-free subgraph with chromatic number κ\kappa?

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Background

Shelah proved that a negative answer is consistent if κ=λ=ℵ1\kappa=\lambda=\aleph_1.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1175 link
  2. [2]Upstream Lean 4 statement for Erdős #1175 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1175 link

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