Erdős (Erdos) Problem #1032: status in 2026

Erdős Problem #1032

We say that a graph is 4-chromatic critical if it has chromatic number 4, and removing any edge decreases the chromatic number to 3. Is there, for arbitrarily large n, a 4-chromatic critical graph on n vertices with minimum degree ≫ n? This is a question of Erdős [Er93, p.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-09-13).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1032 as open (status updated 13 September 2025).
erdosproblems.com lists Erdős problem #917 as open (status updated 31 August 2025).
erdosproblems.com lists Erdős problem #944 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_1032 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1032 on erdosproblems.com	open 2025-09-13	erdosproblems.com
Status of Erdős problem #917 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Status of Erdős problem #944 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_1032`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 2 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-1032,
  title        = {Erdős Problem #1032},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1032}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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-1032.json)](https://cairn-commons.com/problems/erdos-1032)

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

We say that a graph is $4$ -chromatic critical if it has chromatic number $4$ , and removing any edge decreases the chromatic number to $3$ .

Is there, for arbitrarily large $n$ , a $4$ -chromatic critical graph on $n$ vertices with minimum degree $≫ n$ ?

This is a question of Erdős [Er93, p. 341] (see also [Va99, 3.60]). We read $≫ n$ as $\geq c n$ for a fixed constant $c > 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1032».

theorem erdos_1032 :
    ∃ c : ℝ, 0 < c ∧ ∃ᶠ n : ℕ in atTop,
      ∃ G : SimpleGraph (Fin n), IsKCritical G 4 ∧ c * (n : ℝ) ≤ (G.minDegree : ℝ)

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/1032. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_1032.variants.five — Is there, for arbitrarily large n, a 5-chromatic critical graph on n vertices with minimum degree ≫ n?

References

erdosproblems.com/1032
[Er93] Erdős, P., Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. 16 (1993), 333-350.
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).
[Di52] Dirac, G. A., A property of $4$ -chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 27 (1952), 85-92.
[Si72] Simonovits, M., On colour-critical graphs. Studia Sci. Math. Hungar. 7 (1972), 67-81.
[To72] Toft, B., Two theorems on critical $4$ -chromatic graphs. Studia Sci. Math. Hungar. 7 (1972), 83-89.
[Ma15] Markström, K. (ed.), *Problem collection from the IML programme: Graphs, Hypergraphs, and Computing*. arXiv:1511.00270 (2015). Problem 5.1 (M. Simonovits).

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.