Intersection families and Snevily’s conjecture

Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be sets of nonnegative integers with k i > s − r . Let F = { F 1 , F 2 , … , F m } be a family of subsets of [ n ] with | F i | ∈ K for each i and | F i ∩ F j | ∈ L for any i ≠ j . We prove that | F | ≤ ∑ i = s − r s n − 1 i when we have the conditions that | F i | ∉ L and k i ’s are consecutive. We also prove the same bound under the condition ⋂ i = 1 m F i ≠ 0̸ instead of the above conditions. Finally, an observation gives us a bound of n ⌈ n 2 ⌉ on | F | when K ∩ L = 0̸ .