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.