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.