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Level A · Machine-checkable Hard Graph theory P-erdos-151

Erdős Problem #151

For a graph G, let τ(G) denote the minimal number of vertices that include at least one from each maximal clique of G on at least two vertices. Let H(n) be maximal such that every triangle-free graph on n vertices contains an independent set on H(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 #151 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement erdos_151 as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #151 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `erdos_151`research openFormal Conjectures
Formal Conjectures: research statements in the file2 open, 2 solvedFormal 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-151,
  title        = {Erdős Problem #151},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-151}},
  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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On the literature board
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Nobody has worked on this problem here yet

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  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. 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.

or , ,

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

For a graph , let denote the minimal number of vertices that include at least one from each maximal clique of on at least two vertices. Let be maximal such that every triangle-free graph on vertices contains an independent set on vertices.

If is a graph on vertices, is ?

Erdős remarked that this is perhaps completely wrongheaded [Er88].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«151».

theorem erdos_151 :
    ∀ (n : ℕ) (G : SimpleGraph (Fin n)), cliqueTransversalNumber G ≤ n - H n

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/151. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_151.variants.cliqueFree_four — Erdős and Gallai made no progress on the question even when G is K_4-free [Er88].

References

  • erdosproblems.com/151
  • [Er88] Erdős, P., Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.
  • [EGT92] Erdős, P. and Gallai, T. and Tuza, Zs., *Covering the cliques of a graph with vertices*. Discrete Math. (1992), 279-289.

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.