Erdős (Erdos) Problem #761: status in 2026

Erdős Problem #761

The dichromatic number of G, denoted by δ(G), is the minimum number k of colours required such that, in any orientation of the edges of G, there is a k-colouring of the vertices of G such that there are no monochromatic oriented cycles.

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 #761 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #761 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 1 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.

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Cite

@misc{cairn-erdos-761,
  title        = {Erdős Problem #761},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-761}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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

erdos_761.parts.i. The dichromatic number of $G$ , denoted by $δ (G)$ , is the minimum number $k$ of colours required such that, in any orientation of the edges of $G$ , there is a $k$ -colouring of the vertices of $G$ such that there are no monochromatic oriented cycles.

Must a graph with large chromatic number have large dichromatic number?

This question is due to Erdős and Neumann-Lara. We read it for finite graphs: for every $k$ there is $m$ such that every finite graph with $χ (G) \geq m$ has $δ (G) \geq k$ .

erdos_761.parts.ii. The cochromatic number of $G$ , denoted by $ζ (G)$ , is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph.

Must a graph with large cochromatic number contain a graph with large dichromatic number?

This question is due to Erdős and Gimbel. We read it for finite graphs and subgraphs: for every $k$ there is $m$ such that every finite graph with $ζ (G) \geq m$ has a subgraph $H$ with $δ (H) \geq k$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«761» (2 statements).

theorem erdos_761.parts.i :
    ∀ k : ℕ, ∃ m : ℕ, ∀ (V : Type*) [Finite V] (G : SimpleGraph V),
      (m : ℕ∞) ≤ G.chromaticNumber → (k : ℕ∞) ≤ dichromaticNumber G

theorem erdos_761.parts.ii :
    ∀ k : ℕ, ∃ m : ℕ, ∀ (V : Type*) [Finite V] (G : SimpleGraph V),
      (m : ℕ∞) ≤ G.cochromaticNumber →
        ∃ H : G.Subgraph, (k : ℕ∞) ≤ dichromaticNumber H.coe

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/761. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/761
[ErGi93] Erdős, P. and Gimbel, J., Some problems and results in cochromatic theory. Quo vadis, graph theory? (1993), 261-264.

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.