Problem 701

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claims/: The 6 claim pages of Problem 701, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some element $x$ such that whenever $\mathcal{F}'\subseteq \mathcal{F}$ is an intersecting subfamily we have

$$\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.$$

Statement (corrected). Let $\mathcal{F}$ be a family of subsets of a finite set closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some element $x$ such that whenever $\mathcal{F}'\subseteq \mathcal{F}$ is an intersecting subfamily we have

$$\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.$$

Notes. The site's wording names no ground set, and with infinite ground sets and cardinalities it is false. Keith Kearnes's answer of 11 October 2022 to MathOverflow question 432223 takes a cardinal $\kappa$ of countable cofinality above the continuum and builds $\kappa$ countably infinite sets forming an intersecting family whose down-closure has every star of size less than $\kappa$. The problem's discussion thread reached that answer on 22 May 2026, after a counterexample on $\mathbb{N}\times\mathbb{N}$ proposed on 21 May 2026 was shown to be flawed (its larger family is itself intersecting). Every failure needs an infinite family: a finite family closed under taking subsets has only finite members, so its members lie in a finite set.

The change replaces "of sets" with "of subsets of a finite set", in the words of the problem's poser. The site credits the problem to Chvátal, and his Conjecture in [Ch74] (p. 65; library card) is stated for "a family of subsets of a finite set $S$" closed under taking subsets; his Theorem (p. 62), the case of families closed under left shifts, is stated for subsets of $\{1,2,\dots,n\}$. The defect is not in Chvátal's text. Erdős's statement of the conjecture in [Er81], item 4 (p. 26), credits it to Chvátal and also names no ground set, and the site's wording follows that silence. The formal-conjectures statement assumes a finite ground set.

Results about the site's wording are credited here and count for nothing: Kearnes's construction above refutes the wording with infinite ground sets and settles no instance of the corrected Statement. It was not presented as settling the problem, so it has no claim page.

Formulation. The corrected Statement is Chvátal's conjecture, the form the three full claims below prove. A star of the family, the members containing a fixed element, is itself intersecting, so the conjecture says that no intersecting subfamily is larger than the largest star. The reading of the site's wording with infinite ground sets is a variant, and it is false, as the Notes record.

Status. The site labels the problem OPEN (discussion and proof-claims threads accessed 2026-10-07), a label that describes the corrected Statement. The frontmatter's standing follows from three pending full claims of September 2026, all proving the corrected Statement: Chang, Liu and Liu, Keevash and Ellis, Filmus and Friedgut. None is refereed or reviewed by a named outside party, and the site's single proof-claims entry, posted on 30 September 2026 by a forum user for the first preprint and naming the other two, has no comments. The curator's remarks credit partial results of Chvátal [Ch74], Sterboul [St74], Frankl and Kupavskii [FrKu23] and Borg [Bo11], described under Current assessment.

Source. erdosproblems.com/701, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #701, https://www.erdosproblems.com/701.

References.

• [Bo11] Borg, Peter, On Chvátal's conjecture and a conjecture on families of signed sets. European J. Combin. (2011), 140-145.
• [Ch74] Chvátal, V., Intersecting families of edges in hypergraphs having the hereditary property. (1974), 61-66.
• [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42; p. 26.
• [FrKu23] Frankl, Peter and Kupavskii, Andrey, Perfect matchings in down-sets. Discrete Math. (2023), Paper No. 113323, 7.
• [St74] Sterboul, F., Sur une conjecture de V. Chvátal. (1974), 152-164.

Formalization. Statement in formal-conjectures, pinned to the revision described, marked open there with a finite ground set, an undetermined answer and a sorry body.

Current assessment

The site's formulation is Chvátal's conjecture: a family closed under subsets has an element whose star is at least as large as any intersecting subfamily. The corrected Statement adds Chvátal's finite ground set, as the Notes record; it is the statement of Chvátal [Ch74], of the formal-conjectures file and of the three claims.

Three preprints of September 2026 prove the corrected Statement in full, each pending. Chang, Liu and Liu (16 September) prove it through a sharp correlation inequality for increasing Boolean functions; a third-party Lean formalization of their proof is registered at the Palomar registry and linked from their page, and this corpus has not built it. Keevash (20 September) proves Kahn's flow conjecture, a strong form that implies Kleitman's and Chvátal's conjectures. Ellis, Filmus and Friedgut (23 September) give a short spectral proof. The later two credit the first and build on its ideas, as their authors state, and all three disclose AI assistance. None is refereed or reviewed by a named outside party, and the site's label is OPEN, so the problem is claimed with the claim proved by derivation and no claim is accepted.

Three earlier partial results are refereed and have accepted partial claim pages. Frankl and Kupavskii [FrKu23] prove the conjecture for intersecting subfamilies of covering number at most $2$; the site's remark places the covering condition on the whole family, while the paper, as the library card records, places it on the intersecting subfamily. Borg [Bo11] proposed a weighted generalization and proved it for weighted families with a dominant element. Eifler, Gleixner and Pulaj verified the conjecture by an exact integer-programming computation for every family whose members lie in a ground set of at most seven elements. The results of Chvátal [Ch74] and Sterboul [St74], also credited by the site's curator, stay in prose: both appear in the proceedings volume Lecture Notes in Math. 411 (Hypergraph Seminar, Columbus, 1972), and a proceedings volume is not counted as refereed. Chvátal proved the conjecture for families of subsets of $\{1,\dots,n\}$ closed under the stronger compression condition: whenever $A$ is in the family and an injection $f\colon B\to A$ satisfies $x\le f(x)$ for all $x\in B$, then $B$ is in the family. Sterboul proved it when the maximal members all have the same size, pairwise meet in at most one element and at least two of them intersect. A thread comment of 15 May 2026 reports that the covering-number case appears as Theorem 2 of a 1972 working paper of Kleitman and Magnanti, and Eifler, Gleixner and Pulaj state as their Theorem 5, attributed to that working paper, that an intersecting family contained in the union of two stars generates a downset satisfying the conjecture; the published version, J. Combin. Theory Ser. A 16 (1974), 215--220, has no claim page because its statement has not been compiled here. A thread comment of 9 May 2026 gives a proof for families of rank at most $2$; it is a thread post, not a dated manuscript, and has no page.

Search scope, 2026-10-07: the site's page, its discussion thread (six comments) and proof-claims thread (one entry), the community database (teorth/erdosproblems, formalized statement recorded), the formal-conjectures statement file, the arXiv records of the three preprints and of the papers of Frankl and Kupavskii and of Eifler, Gleixner and Pulaj, the zbMATH review of [Bo11] and the Palomar registry. Remaining gaps: no refereed publication or independent review of any of the three full proofs is recorded; the site's wording, read with infinite ground sets, is false, as the Notes record.

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.

• chvatal_1974_intersecting_families_edges_hypergraphs_hereditary_property
• chvatal_1974_intersecting_families_edges_hypergraphs_hereditary_property / conjecture_p65
• chvatal_1974_intersecting_families_edges_hypergraphs_hereditary_property / theorem_p62
• frankl_2023_perfect_matchings_down_sets
• frankl_2023_perfect_matchings_down_sets / theorem_4
• frankl_2023_perfect_matchings_down_sets / theorem_5
• frankl_2023_perfect_matchings_down_sets / theorem_6
• frankl_2023_perfect_matchings_down_sets / theorem_7