Chvátal's intersecting-family conjecture asserts that a family
of subsets of a nonempty finiteset, closed under taking subsets, has
a largest intersecting family consisting of
all its members that contain one fixed element (Chvátal 1974). More precisely,
for every intersecting family,
Bloom, T. F. "Erdős Problem 701." Accessed Oct. 26, 2026. https://www.erdosproblems.com/701.Chang,
F.; Liu, H.; and Liu, M. "A Proof of Chvátal's Conjecture via a Sharp
Correlation Inequality." 16 Sep 2026. https://arxiv.org/abs/2609.19123.Chvátal,
V. "Intersecting Families of Edges in Hypergraphs Having the Hereditary Property."
In Hypergraph Seminar (Ed. C. Berge and D. Ray-Chaudhuri). Berlin,
Germany: Springer-Verlag, pp. 61-66, 1974. https://doi.org/10.1007/BFb0066179.Keevash,
P. "On Kahn's Flow Conjecture." 20 Sep 2026. https://arxiv.org/abs/2609.23595.
of subsets of a nonempty finite set 
, closed under taking subsets, has
a largest intersecting family consisting of
all its members that contain one fixed element (Chvátal 1974). More precisely,
for every intersecting family 
,
