On the number of edges in a uniform hypergraph with a range of permitted intersections

The paper studies the quantity p ( n , k , t 1 , t 2 ) equal to the maximum number of edges in a k -uniform hypergraph with the property that the size of the intersection of any two edges lies in the interval [ t 1 , t 2 ]. Previously known upper and lower bounds are given. New bounds for p ( n , k , t 1 , t 2 ) are obtained, and the relationship between these bounds and known estimates is studied. For some parameter values, the exact values of p ( n , k , t 1 , t 2 ) are explicitly calculated.