Set systems with -intersections modulo a prime number

Let p be a prime and let L = { l 1 , l 2 , … , l s } and K = { k 1 , k 2 , … , k r } be two subsets of { 0 , 1 , 2 , … , p − 1 } satisfying max l j < min k i . We will prove the following results: If F = { F 1 , F 2 , … , F m } is a family of subsets of [ n ] = { 1 , 2 , … , n } such that | F i ∩ F j | ( mod p ) ∈ L for every pair i ≠ j and | F i | ( mod p ) ∈ K for every 1 ⩽ i ⩽ m , then | F | ⩽ ( n − 1 s ) + ( n − 1 s − 1 ) + ⋯ + ( n − 1 s − 2 r + 1 ) . If either K is a set of r consecutive integers or L = { 1 , 2 , … , s } , then | F | ⩽ ( n − 1 s ) + ( n − 1 s − 1 ) + ⋯ + ( n − 1 s − r ) . We will also prove similar results which involve two families of subsets of [ n ] . These results improve the existing upper bounds substantially.