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Level A · Machine-checkable Hard Graph theory P-erdos-596

Erdős Problem #596

Erdős Problem 596 (Erdős–Hajnal, [Er87]). For which graph pairs (G_1, G_2) is it true that (1) for every n ≥ 1 there is a graph H without a G_1 such that any n-colouring of H's edges contains a monochromatic G_2, and yet (2) for every graph H without a G_1 there is an aleph_0-colouring of H's edges…

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-596,
  title        = {Erdős Problem #596},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-596}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Current state

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

The problem

The question

Erdős Problem 596 (Erdős–Hajnal, [Er87]). For which graph pairs is it true that

(1) for every there is a graph without a such that any -colouring of 's edges contains a monochromatic , and yet (2) for every graph without a there is an -colouring of 's edges with no monochromatic ?

Erdős and Hajnal originally conjectured that no such pair exists; but witnesses it (Nešetřil–Rödl + Erdős–Hajnal). The full question is to characterise the class of all such pairs, recorded here as answer(sorry).

See Problem 595 for the specific case .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«596». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_596 :
    ∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),
      IsErdosHajnalExceptional G₁ G₂ ↔
      (answer(sorry) : ∀ {U₁ U₂ : Type}, SimpleGraph U₁ → SimpleGraph U₂ → Prop) G₁ G₂

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/596. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_596.variants.K4_K3_exceptional_iff — Whether (K_4, K_3) is Erdős–Hajnal exceptional is precisely the content of Erdős Problem 595.

References

  • erdosproblems.com/596
  • [Er87] Erdős, Some of my favourite problems in various branches of combinatorics, Mat. Lapok 1987.
  • [NeRo75] Nešetřil and Rödl, *The Ramsey property for graphs with forbidden complete subgraphs*, J. Combin. Theory B 20 (1976), 243--249.

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.