Erdős (Erdos) Problem #813: status in 2026

Erdős Problem #813

Let h(n) be minimal such that every graph on n vertices where every set of 7 vertices contains a triangle must contain a clique on at least h(n) vertices. Estimate h(n). In particular, do there exist constants c_1, c_2 > 0 such that n^1/3+c_1 ≪ h(n) ≪ n^1/2-c_2?

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 #813 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_813 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #813 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_813`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 3 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-813,
  title        = {Erdős Problem #813},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-813}},
  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

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

Let $h (n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle must contain a clique on at least $h (n)$ vertices. Estimate $h (n)$ . In particular, do there exist constants $c_{1}, c_{2} > 0$ such that $n^{1/3 + c_{1}} ≪ h (n) ≪ n^{1/2 - c_{2}} ?$

The lower half has a positive answer (Erdos813.erdos_813.variants.lower); the upper half is Erdos813.erdos_813.variants.upper.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«813».

theorem erdos_813 :
    ∃ c₁ > (0 : ℝ), ∃ c₂ > (0 : ℝ),
      (fun n : ℕ ↦ (n : ℝ) ^ (1 / 3 + c₁ : ℝ)) =O[atTop] (fun n ↦ (h n : ℝ)) ∧
        (fun n : ℕ ↦ (h n : ℝ)) =O[atTop] (fun n : ℕ ↦ (n : ℝ) ^ (1 / 2 - c₂ : ℝ))

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/813. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_813.variants.upper — The upper half of Erdos813.erdos_813: is there a constant c_2 > 0 with h(n) ≪ n^1/2-c_2?

References

erdosproblems.com/813
[Er91] Erdős, P., *Problems and results in combinatorial analysis and combinatorial number theory*. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[BuSu23] Bucić, M. and Sudakov, B., Large independent sets from local considerations. Combinatorica 43 (2023), 505-546. arXiv:2007.03667

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.