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Statement

Conjecture (p. 65, unnumbered), as posed: "Let $F$ be a family of subsets of a finite set $S$ such that $X\in F$, $Y\subset X\Rightarrow Y\in F$. Then there is a $t\in S$ such that every intersecting subfamily $G$ of $F$ satisfies $|G|\le|\{X\in F: t\in X\}|$."

Here a family is intersecting when any two of its members, not necessarily distinct, meet (p. 62). The hypothesis is closure under taking subsets, and the finite ground set $S$ is part of it. The paper introduces the conjecture as a possible "strengthening of our theorem" (p. 65): the Theorem (p. 62) gives the conclusion with $t=1$ under the stronger hypothesis of closure under the left-shift order on $S=\{1,\dots,n\}$.

Source. V. Chvátal, Intersecting families of edges in hypergraphs having the hereditary property, in: Hypergraph Seminar (Ohio State Univ., Columbus, 1972), Lecture Notes in Math. 411, Springer, Berlin, 1974, pp. 61--66; the Conjecture on p. 65. The edition is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image, and the quotation matches the print. A conjecture has no proof to check.

Proof pointer

None; the paper poses the statement as a conjecture and proves only the case of the Theorem (p. 62).

Dependencies

None.

Bears on

• Problem 701: this Conjecture is the problem's corrected Statement, which restores the finite ground set $S$ that the site's wording omits. The paper records no proof of it; the claims about it are recorded on the problem page.