Erdős Problem #1176 — open problem in logic & formalisation

Erdős Problem #1176

Let G be a graph with chromatic number aleph_1. Is it true that there is a colouring of the edges with aleph_1 many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours? A problem of Erdős, Galvin, and Hajnal.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-1176,
  title        = {Erdős Problem #1176},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1176}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $G$ be a graph with chromatic number $ℵ_{1}$ . Is it true that there is a colouring of the edges with $ℵ_{1}$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?

A problem of Erdős, Galvin, and Hajnal. The consistency of this was proved by Hajnal and Komjáth.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1176». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1176 :
    answer(sorry) ↔ ∀ {V : Type*} (G : SimpleGraph V), G.chromaticCardinal = aleph 1 →
      ∃ (EColor : Type) (_ : mk EColor = aleph 1) (c_edge : G.edgeSet → EColor),
        ∀ (VColor : Type) (_ : mk VColor ≤ aleph 0) (c_vert : V → VColor),
          ∃ (vc : VColor),
            ∀ (ec : EColor), ∃ (u v : V) (h : G.Adj u v),
              c_vert u = vc ∧ c_vert v = vc ∧ c_edge ⟨s(u, v), h⟩ = ec

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1176. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1176

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.