Erdős (Erdos) Problem #500: status in 2026

Erdős Problem #500

What is ex_3(n,K_4^3)? That is, the largest number of 3-edges which can placed on n vertices so that there exists no K_4^3, a set of 4 vertices which is covered by all 4 possible 3-edges. See also 712 for the general case.

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

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #500 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_500 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #500 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_500`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 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. 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-500,
  title        = {Erdős Problem #500},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-500}},
  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

What is $ex_{3} (n, K_{4}^{3})$ ? That is, the largest number of $3$ -edges which can placed on $n$ vertices so that there exists no $K_{4}^{3}$ , a set of 4 vertices which is covered by all 4 possible $3$ -edges.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«500».

theorem erdos_500 :
    (fun n ↦ Hypergraph.cliqueExtremalNumber n 3 4) = erdos_500.answer

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/500. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_500.variants.turan_conjecture — Turán observed that dividing the vertices into three equal parts X_1,X_2,X_3, and taking the edges to be those triples that either have exactly one vertex in…
erdos_500.variants.asymptotic — The asymptotic form of the question: what is the Turán density π(K_4^3)=lim_n→inftyex_3(n,K_4^3)/C(n, 3)? By erdos_500.variants.lower_bound, π(K_4^3)≥5/9.

References

erdosproblems.com/500
[Ra10] Razborov, Alexander A., On 3-hypergraphs with forbidden 4-vertex configurations. SIAM J. Discrete Math. (2010), 946-963.
[BaTa11] Baber, Rahil and Talbot, John, Hypergraphs do jump. Combin. Probab. Comput. (2011), 161-171.
[Ba12] Baber, Rahil, Turán densities of hypercubes. arXiv:1201.3587 (2012).
[JPILY26] Jeong, Gyeongwon and Park, Seonghun and Im, Seonghyuk and Lee, Joonkyung and Yang, Hongseok, A New Upper Bound for the Turán Density of the Tetrahedron. arXiv:2609.27495 (2026).

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.