Erdős (Erdos) Problem #111: status in 2026

Erdős Problem #111

Erdős Problem 111 (Erdős, Hajnal, Szemerédi [EHS82]): If G is a graph with chromatic number aleph_1, is it true that h_G(n)/n → ∞?

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 #111 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_111 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #111 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_111`	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.

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Cite

@misc{cairn-erdos-111,
  title        = {Erdős Problem #111},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-111}},
  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

Erdős Problem 111 (Erdős, Hajnal, Szemerédi [EHS82]):

If $G$ is a graph with chromatic number $ℵ_{1}$ , is it true that $h_{G} (n) / n \to \infty$ ?

Formally: for every graph $G$ (on a type $V$ ) with chromatic cardinal $ℵ_{1}$ , the ratio $h_{G} (n) / n$ tends to infinity: for every constant $C$ , all sufficiently large $n$ satisfy $h_{G} (n) > C \cdot n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«111».

theorem erdos_111 :
    ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
      Filter.Tendsto (fun n : ℕ => (hG G n : ℝ) / n) Filter.atTop Filter.atTop

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/111. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_111.variants.erdos_conjecture — Erdős conjectured [Er81] that for every ε > 0 there is a graph G with χ(G) = aleph_1 such that h_G(n) ≪ n^1+ε. The graph is chosen after ε.

References

erdosproblems.com/111
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. (1981)
[EHS82] Erdős, P., Hajnal, A., and Szemerédi, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117–123.
[Er87] Erdős, P., Some problems on finite and infinite graphs. (1987)
[Er90] Erdős, P., Some of my favourite unsolved problems. (1990)
[Er97d] Erdős, P., Some recent problems and results in graph theory. (1997)
[Er97f] Erdős, P., Some unsolved problems. (1997)

Formalization notes

bipartiteDistance G: the minimum number of edges to delete from a (finite) simple graph $G$ to make it bipartite. Defined as the infimum over all spanning subgraphs $H \leq G$ with H.IsBipartite of the number of edges in G.edgeSet \ H.edgeSet.

hG G n: for a graph $G$ on vertex type $V$ , hG G n is the supremum over all sets $S$ of vertices of cardinality $n$ of bipartiteDistance (G.induce S). This matches the problem's formulation: "any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_{G} (n)$ edges."

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.