Chvátal’s hereditary-family conjecture

Prove that every hereditary family of sets has an element-centered intersecting subfamily at least as large as every intersecting subfamily of the hereditary family.

Background

Chvátal’s conjecture asserts a broad extension of the Erdős–Ko–Rado phenomenon from uniform set systems to arbitrary hereditary families. The survey defines a family as centered if one element belongs to at least as many members as the size of any intersecting subfamily.

The conjecture was posed in 1972 and is described as still difficult, with only several special cases resolved.

References

It is a monument to the difficulty of the conjecture that it is so well-known and yet so few papers have addressed it in the half century since its birth.

— A Survey of the Holroyd-Talbot Conjecture  (2501.16144 - Hurlbert, 27 Jan 2025) in Conjecture 2, Section 4