Resolve the modified uniform Erdős–Frankl–Pach conjecture

Establish whether every set family \(\mathcal{F}\subseteq\binom{[n]}{d+1}\) satisfying, for each \(F\in\mathcal{F}\), the existence of an \(s\)-element subset \(B_F\subseteq F\) such that \(F\cap F'\ne B_F\) for every \(F'\in\mathcal{F}\), has size at most \(\binom{n-1}{d}\), for all \(n\ge 2(d+1)\) and \(0\le s\le d\).

Background

The paper introduces a refined conjecture intended to recover a robust connection between VC-dimension and the Erdős–Ko–Rado theorem after the original Erdős–Frankl–Pach conjecture was disproved. It proves the conjecture for s\in{0,d} and for s=1 when n is sufficiently large relative to d, but leaves the general case unresolved.