Developing ideas of Blokhuis, the authors generalize the Ray-Chaudhuri- Wilson theorem to show: If \(f\) is a family of subsets of an \(n\) element set such that for any distinct \(E,F\in f\), we have \(| E|\), \(| F|\in K\) and \(| E\cap F|\in L\), then \(| f|\leq{n\choose| L|}+{n\choose| L- 1|}+\cdots+{n\choose\max\{0,| L|-| K|+1\}}\). They also generalize the Frankl-Wilson theorem, which essentially gives a version of the above result where we now consider the elements of \(K\) and \(L\) modulo \(p\).