Set systems with LL-intersections modulo a prime number

Let p   be a prime and let L={l1,l2,…,ls}L={l1,l2,…,ls} and K={k1,k2,…,kr}K={k1,k2,…,kr} be two subsets of {0,1,2,…,p−1}{0,1,2,…,p−1} satisfying maxlj