How can I determine the size of the largest collection of k-element subsets of an...

Let $L$ be a set of m integers and $F$ be an $L$-intersecting $k$-uniform family of subsets of a set of $n$ elements, where $m\le k$, then
$|F|\le (\genfrac{}{}{0ex}{}{n}{m})$
$k$-uniform family is a set of subsets, each subset being of size $k$.
An $L$-intersecting family is such that the intersection size of any two distinct sets in the family is in $L$.
For every $k\ge m\ge 1$ and $n\ge 2k^2$ there exists a $л$-uniform family $F$ of size $>(\frac{n}{2k}{)}^m$ on $n$ points such that $|A\cap B|\le m-1$ for any two distinct sets $A,B\in F$.