Erdős (Erdos) Problem #596: status in 2026

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…

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #596 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_596 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #596 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_596`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 6 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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.

Try it with your chatbot Submit a claim Follow (RSS)

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 2026-11-13}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Ferdos-596.json)](https://cairn-commons.com/problems/erdos-596)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Erdős Problem 596 (Erdős–Hajnal, [Er87]). For which graph pairs $(G_{1}, G_{2})$ is it true that

(1) for every $n \geq 1$ 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 $ℵ_{0}$ -colouring of $H$ 's edges with no monochromatic $G_{2}$ ?

Erdős and Hajnal originally conjectured that no such pair exists; but $(C_{4}, C_{6})$ witnesses it (Nešetřil–Rödl + Erdős–Hajnal). The full question is to characterise the class of all such pairs, recorded here as the unknown erdos_596.answer.

See Problem 595 for the specific case $(G_{1}, G_{2}) = (K_{4}, K_{3})$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«596».

theorem erdos_596 :
    ∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),
      IsErdosHajnalExceptional G₁ G₂ ↔ erdos_596.answer 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.