Set systems with I-intersections modulo a prime number

Let p be a prime and let L = {l1, l2, . . . , ls} and K = {k1, k2, . . . , kr} be two subsets of {0, 1, 2, . . . , p − 1} satisfying max lj < min ki. We will prove the following results: If F = {F1, F2, . . . , Fm} is a family of subsets of [n] = {1, 2, . . . , n} such that |Fi ∩ Fj | (mod p) ∈ L for every pair i 6= j and |Fi| (mod p) ∈ K for every 1 ≤ i ≤ m, then |F| ≤ ( n− 1 s ) + ( n− 1 s− 1 ) + · · ·+ ( n− 1 s− 2r + 1 ) .