Erdős #1068 · graph theory / set theory
Does every graph with chromatic number contain a countable subgraph which is infinitely vertex-connected?
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Problem statement
Does every graph with chromatic number contain a countable subgraph which is infinitely vertex-connected?
Lean 4
Formal statement
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- Upstream Lean 4 statement for Erdős #1068 ↗
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- Formal Conjectures — Erdős Problem 1068 ↗
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Background
This is described in [BoPi24] as a 'version of the Erdős-Hajnal problem' (which is [1067]), but it does not seem to appear in [ErHa66].
We say a graph is infinitely (vertex) connected if any two vertices are connected by infinitely many pairwise vertex-disjoint paths.
Soukup [So15] constructed a graph with uncountable chromatic number in which every uncountable set is finitely vertex-connected. A simpler construction was given by Bowler and Pitz [BoPi24].
See also [1067].
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.
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