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OP-9084OpenLogic and foundations

Erdős #593 · set theory / graph theory

Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number >ℵ0>\aleph_0.

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

Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number >ℵ0>\aleph_0.

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Background

Similar problems were investigated by Erdős, Galvin, and Hajnal [EGH75]. Erdős claims that for graphs the problem is completely solved: a graph of chromatic number ≥ℵ1\geq \aleph_1 must contain all finite bipartite graphs but need not contain any fixed odd cycle.

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 #593 link
  2. [2]Upstream Lean 4 statement for Erdős #593 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 593 link

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