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Level A · Machine-checkable Hard Graph theory P-erdos-918

Erdős Problem #918

Erdős Problem #918

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.

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Cite 
@misc{cairn-erdos-918,
  title        = {Erdős Problem #918},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-918}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

erdos_918.parts.i. (no description upstream)

erdos_918.parts.ii. (no description upstream)

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«918» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_918.parts.i :
    answer(sorry) ↔ ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧
      ∀ (W : Set V) (_ : #W = ℵ₁), (G.induce W).chromaticCardinal ≤ ℵ₀
theorem erdos_918.parts.ii :
    answer(sorry) ↔
    ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
      ∀ (W : Set V) (_ : #W = ℵ_ ω), (G.induce W).chromaticCardinal ≤ ℵ₀

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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/918. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_918.variants.all_subgraphs.parts.i
  • erdos_918.variants.all_subgraphs.parts.ii — Is there a graph with aleph_ω+1 vertices and chromatic number aleph_1 such that every subgraph on aleph_ω vertices has chromatic number ≤aleph_0?

References

  • erdosproblems.com/918
  • [ErHa68b] Erdős, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.
  • [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.

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.