Erdős (Erdos) Problem #1111: status in 2026

Erdős Problem #1111

If G is a finite graph and A,B are disjoint sets of vertices then we call A,B anticomplete if there are no edges between A and B. If t,c≥ 1 then there exists d≥ 1 such that if χ(G)≥ d and ω(G)<t then there are anticomplete sets A,B with χ(A)≥ χ(B)≥ c.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-12-07).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1111 as open (status updated 7 December 2025).
Formal Conjectures marks the Lean statement erdos_1111 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1111 on erdosproblems.com	open 2025-12-07	erdosproblems.com
Formal Conjectures: status of `erdos_1111`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 4 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.

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Cite

@misc{cairn-erdos-1111,
  title        = {Erdős Problem #1111},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1111}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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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

If $G$ is a finite graph and $A, B$ are disjoint sets of vertices then we call $A, B$ anticomplete if there are no edges between $A$ and $B$ .

If $t, c \geq 1$ then there exists $d \geq 1$ such that if $χ (G) \geq d$ and $ω (G) < t$ then there are anticomplete sets $A, B$ with $χ (A) \geq χ (B) \geq c$ .

El-Zahar and Erdős [ElEr85] pose this as a question. Here $χ (A)$ is the chromatic number of the induced subgraph $G [A]$ . Swapping $A$ and $B$ , the conclusion says that both $χ (A)$ and $χ (B)$ are at least $c$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1111».

theorem erdos_1111 :
    ∀ t c : ℕ, 1 ≤ t → 1 ≤ c → ∃ d : ℕ, 1 ≤ d ∧
      ∀ (V : Type*) [Finite V] (G : SimpleGraph V),
        (d : ℕ∞) ≤ G.chromaticNumber → G.CliqueFree t →
          ∃ A B : Set V, Anticomplete G A B ∧
            (c : ℕ∞) ≤ (G.induce A).chromaticNumber ∧
            (c : ℕ∞) ≤ (G.induce B).chromaticNumber

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/1111. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1111
[ElEr85] El-Zahar, M. and Erdős, P., *On the existence of two non-neighboring subgraphs in a graph*. Combinatorica 5 (1985), 295-300.
[NSS24] Nguyen, T., Scott, A. and Seymour, P., On a problem of El-Zahar and Erdős. J. Combin. Theory Ser. B 165 (2024), 211-222. arXiv:2303.13449
[Wa80b] Wagon, S., *A bound on the chromatic number of graphs without certain induced subgraphs*. J. Combin. Theory Ser. B 29 (1980), 345-346.

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.