Maximum intersection of k-neighborhoods of a family of n-subsets

Question

Given a family $\mathcal{F}$ of distinct $(n-1)$-subsets of a ground set $S$ on $N$ elements (where $N > 2n$), what is the maximum cardinality of a family $\mathcal{F}'$ of $n$-subsets of $S$, such that $|A \cap B| \geq k$ (where $1 \leq k \leq n-1$) whenever $A \in \mathcal{F}$ and $B \in \mathcal{F}'$? I am looking for a non trivial lower bound for $\frac{{N \choose n} - |\mathcal{F}'|}{|\mathcal{F}|}$ over all $\mathcal{F}$ for a fixed (N, n, k)-tuple. Pointers to variants of the problem or known relevant theorems might help too.