Almost entropic matroids and $(k,L)$-systems

For an integer $k$ and a set of integers $L$, a $(k,L)$-system is a family of $k$-element sets such that the size of the intersection of any two distinct members belongs to $L$. Let $m(n,k,L)$ denote the maximum size of a $(k,L)$-system on a ground set of $n$ elements. In this work, we establish a close connection between the asymptotic growth of $m(n,k,L)$ and almost entropic matroids. As a consequence, we obtain a general computable upper bound on the asymptotics of $m(n,k,L)$ that unifies and strengthens all previously known upper bounds.