Erdős (Erdos) Problem #611: status in 2026

Erdős Problem #611

For a graph G let τ(G) denote the minimal number of vertices that include at least one from each maximal clique of G. Is it true that if all maximal cliques in G have at least cn vertices then τ(G) = o_c(n)? Here n is the number of vertices, and o_c(n) is uniform over all such graphs on n vertices.

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

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #611 on erdosproblems.com	open 2025-08-31	erdosproblems.com
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.

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Cite

@misc{cairn-erdos-611,
  title        = {Erdős Problem #611},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-611}},
  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_611.parts.i. For a graph $G$ let $τ (G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ . Is it true that if all maximal cliques in $G$ have at least $c n$ vertices then $τ (G) = o_{c} (n)$ ?

Here $n$ is the number of vertices, and $o_{c} (n)$ is uniform over all such graphs on $n$ vertices. $τ (G)$ is SimpleGraph.cliqueTransversalNumber, which (as in Erdős Problem 151) ignores maximal cliques on one vertex; this makes no difference here, since for large $n$ every maximal clique has at least $c n \geq 2$ vertices.

erdos_611.parts.ii. For $0 < c < 1$ , estimate the minimal $k_{c} (n)$ such that if every maximal clique in a graph $G$ on $n$ vertices has at least $k_{c} (n)$ vertices then $τ (G) < (1 - c) n$ .

We require $c < 1$ : for $c \geq 1$ the bound $τ (G) < (1 - c) n \leq 0$ is impossible, so $k_{c} (n)$ is degenerate. Erdős, Gallai, and Tuza [EGT92] proved a lower bound of the form $k_{c} (n) \geq n^{c^{'} / l o g l o g n}$ for some $c^{'} > 0$ (as reported on erdosproblems.com).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«611» (2 statements).

theorem erdos_611.parts.i :
    ∀ c > (0 : ℝ), ∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop, ∀ G : SimpleGraph (Fin n),
      AllMaximalCliquesAtLeast G (c * (n : ℝ)) →
        (G.cliqueTransversalNumber : ℝ) ≤ ε * (n : ℝ)

theorem erdos_611.parts.ii :
    ∀ c : ℝ, 0 < c → c < 1 →
      (fun n ↦ (cliqueThreshold c n : ℝ)) =Θ[atTop] erdos_611.parts.ii.answer c

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/611. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/611
[EGT92] Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, *Covering the cliques of a graph with vertices*. Discrete Math. (1992), 279-289.
[Er94] Erdős, P., Problems and results on set systems and hypergraphs. Extremal problems for finite sets (Visegrád, 1991) (1994), 217-227.
[Er99] Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6.

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.