Problem 643

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claims/: The 1 claim page of Problem 643, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $t\geq 3$

$$f(n;t)=(1+o(1))\binom{n}{t-1}?$$

Status. Open.

Source. erdosproblems.com/643, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #643, https://www.erdosproblems.com/643.

References.

• [Fu84] Füredi, Z., Hypergraphs in which all disjoint pairs have distinct unions. Combinatorica (1984), 161-168.
• [PiVe09] Pikhurko, Oleg and Verstraëte, Jacques, The maximum size of hypergraphs without generalized 4-cycles. J. Combin. Theory Ser. A (2009), 637-649.

Formalization. Statement in formal-conjectures.

Current assessment

The status above is the site's label. The site's commentary records Füredi's bounds $\binom{n-1}{t-1}+\lfloor(n-1)/t\rfloor\le f(n;t)<\frac72\binom{n}{t-1}$, his conjecture that the lower bound is sharp for $t\ge4$, and the upper bounds of Pikhurko and Verstraëte. One result is claimed from outside the project: Huang, Ma and Yang's preprint of 2026-09-29, linked from the site's discussion thread, proves Füredi's conjecture for every fixed $t\ge4$ and large $n$, which answers the asymptotic question yes for those $t$; it is a partial claim, not refereed and not accepted by the site, and the case $t=3$ stays open, so the standing derived in the frontmatter is open. This page records no literature search beyond the site and no independent assessment of proof coverage.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• furedi_1984_hypergraphs_which_all_disjoint_pairs_have
• furedi_1984_hypergraphs_which_all_disjoint_pairs_have / conjecture_1_4
• furedi_1984_hypergraphs_which_all_disjoint_pairs_have / example_1_3
• furedi_1984_hypergraphs_which_all_disjoint_pairs_have / lemma_3_3
• furedi_1984_hypergraphs_which_all_disjoint_pairs_have / proposition_6_1
• furedi_1984_hypergraphs_which_all_disjoint_pairs_have / theorem_1_2
• pikhurko_2009_maximum_size_hypergraphs_without_generalized_4
• pikhurko_2009_maximum_size_hypergraphs_without_generalized_4 / lemma_3
• pikhurko_2009_maximum_size_hypergraphs_without_generalized_4 / theorem_1
• pikhurko_2009_maximum_size_hypergraphs_without_generalized_4 / theorem_2