A hypergraph ℋ is an (n, m)-hypergraph if it contains n vertices and m hyperedges, where n ≥ 1 and m ≥ 0 are two integers. Let k be a positive integer and let L be a set of nonnegative integers. A hypergraph ℋ is k-uniform if all its hyperedges have the same size k, and ℋ is L-intersecting if the number of common vertices of every two hyperedges belongs to L. In this paper, we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting (n, m)-hypergraphs for fixed n, m and L. We will provide some tight upper and lower bounds on k in terms of n, m and L. [ABSTRACT FROM AUTHOR]