On the Number of Edges of a Uniform Hypergraph with a Range of Allowed Intersections

We study the quantity p(n, k, t(1), t(2)) equal to the maximum number of edges in a k-uniform hypergraph having the property that all cardinalities of pairwise intersections of edges lie in the interval [t(1), t(2)]. We present previously known upper and lower bounds on this quantity and analyze their interrelations. We obtain new bounds on p(n, k, t(1), t(2)) and consider their possible applications in combinatorial geometry problems. For some values of the parameters we explicitly evaluate the quantity in question. We also give a new bound on the size of a constant-weight error-correcting code.