Erdős (Erdos) Problem #712: status in 2026

Erdős Problem #712

Determine, for any k>r>2, the limiting value as n→∞ of ex_r(n,K_k^r)/C(n, r), where ex_r(n,K_k^r) is the largest number of r-edges which can placed on n vertices so that there exists no set of k vertices which is covered by all C(k, r) possible r-edges.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $500.

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-712,
  title        = {Erdős Problem #712},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-712}},
  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

Determine, for any $k > r > 2$ , the limiting value as $n \to \infty$ of $\frac{ex _{r} ( n , K _{k}^{r} )}{( r n )},$ where $ex_{r} (n, K_{k}^{r})$ is the largest number of $r$ -edges which can placed on $n$ vertices so that there exists no set of $k$ vertices which is covered by all $(r k)$ possible $r$ -edges.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«712».

theorem erdos_712 :
    ∀ k r : ℕ, 2 < r → r < k →
      Tendsto (fun n ↦ (Hypergraph.cliqueExtremalNumber n r k : ℝ) / n.choose r)
        atTop (𝓝 (erdos_712.answer k 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/712. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/712
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.

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.