Erdős (Erdos) Problem #643: status in 2026

Erdős Problem #643

Let f(n;t) be minimal such that if a t-uniform hypergraph on n vertices contains at least f(n;t) edges then there must be four edges A,B,C,D such that A∪ B= C∪ D and A∩ B=C∩ D=∅. Estimate f(n;t) - in particular, is it true that for t≥ 3 f(n;t)=(1+o(1))C(n, t-1)?

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 #643 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #643 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 1 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.

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Cite

@misc{cairn-erdos-643,
  title        = {Erdős Problem #643},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-643}},
  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

erdos_643.parts.i. Let $f (n; t)$ be minimal such that if a $t$ -uniform hypergraph on $n$ vertices contains at least $f (n; t)$ edges then there must be four edges $A, B, C, D$ such that $A \cup B = C \cup D$ and $A \cap B = C \cap D = \emptyset.$ Estimate $f (n; t)$ - in particular, is it true that for $t \geq 3$ $f (n; t) = (1 + o (1)) (t - 1 n) ?$

erdos_643.parts.ii. Is it true that $f (n; t) = (1 + o (1)) (t - 1 n)$ for $t \geq 3$ ?

For $t \geq 4$ , Huang, Ma, and Yang [HMY26] prove the exact extremal formula below. The case $t = 3$ remains open.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«643» (2 statements).

theorem erdos_643.parts.i :
    ∀ t : ℕ, 1 ≤ t →
      (fun n ↦ (forcingThreshold n t : ℝ)) ~[atTop] erdos_643.parts.i.answer t

theorem erdos_643.parts.ii :
    ∀ t : ℕ, 3 ≤ t →
    (fun n ↦ (forcingThreshold n t : ℝ)) ~[atTop]
        (fun n ↦ (n.choose (t - 1) : ℝ))

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/643. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/643
[Fu84] F\"uredi, Z., Hypergraphs in which all disjoint pairs have distinct unions. Combinatorica (1984), 161--168.
[PiVe09] Pikhurko, Oleg and Verstra\"{e}te, Jacques, The maximum size of hypergraphs without generalized 4-cycles. J. Combin. Theory Ser. A (2009), 637--649.
[HMY26] Huang, Hao, Ma, Jie, and Yang, Tianchi, *Extremal hypergraphs without generalized 4-cycles*. arXiv:2609.37744 (2026), Theorem 1.2.

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.