Erdős (Erdos) Problem #719: status in 2026

Erdős Problem #719

Let r≥ 2, and let ex_r(n;K_r+1^r) be the maximum number of r-edges that can be placed on n vertices without forming a K_r+1^r (the r-uniform complete graph on r+1 vertices).

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

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-719,
  title        = {Erdős Problem #719},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-719}},
  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 $r \geq 2$ , and let $ex_{r} (n; K_{r + 1}^{r})$ be the maximum number of $r$ -edges that can be placed on $n$ vertices without forming a $K_{r + 1}^{r}$ (the $r$ -uniform complete graph on $r + 1$ vertices). Is every $r$ -hypergraph $G$ on $n$ vertices the union of at most $ex_{r} (n; K_{r + 1}^{r})$ many copies of $K_{r}^{r}$ and $K_{r + 1}^{r}$ , no two of which share a $K_{r}^{r}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«719».

theorem erdos_719 :
    ∀ r : ℕ, 2 ≤ r → ∀ (n : ℕ) (H : Finset (Finset (Fin n))),
    H.IsUniform r → ∃ D : Finset (Finset (Fin n)),
      H.IsCompleteHypergraphDecomposition r D ∧
        D.card ≤ Hypergraph.cliqueExtremalNumber n r (r + 1)

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/719. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/719

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.