Intersection families and Snevily’s conjecture

Let K={k1,k2,…,kr}K={k1,k2,…,kr} and L={l1,l2,…,ls}L={l1,l2,…,ls} be sets of nonnegative integers with ki>s−rki>s−r. Let F={F1,F2,…,Fm}F={F1,F2,…,Fm} be a family of subsets of [n][n] with |Fi|∈K|Fi|∈K for each ii and |Fi∩Fj|∈L|Fi∩Fj|∈L for any i≠ji≠j. We prove that |F|≤∑i=s−rsn−1i when we have the conditions that |Fi|∉L|Fi|∉L and kiki’s are consecutive. We also prove the same bound under the condition ⋂i=1mFi≠0̸ instead of the above conditions. Finally, an observation gives us a bound of n⌈n2⌉ on |F||F| when K∩L=0̸K∩L=0̸.