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Chvátal's Intersecting-Family Conjecture

Chvátal's intersecting-family conjecture asserts that a family D of subsets of a nonempty finite set X, 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 I subset= D,

|I|<=max_(x in X)|{A in D:x in A}|.

Empty sets are excluded from the intersecting family.

Chang et al. (2026) and Keevash (2026) reported proofs. Keevash's approach proves the stronger Kahn's flow conjecture.

This is problem 701 in the Erdős problems collection (Bloom 2026).

See also

Erdős Problems, Intersecting Family, Kahn's Flow Conjecture

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References

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.

Cite this as:

Weisstein, Eric W. "Chvátal's Intersecting-Family Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChvatalsIntersecting-FamilyConjecture.html

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