DOIAbstract
AbstractLet 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 maxlj<minki. 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|(modp)∈L for every pair i≠j and |Fi|(modp)∈K for every 1⩽i⩽m, then|F|⩽(n−1s)+(n−1s−1)+⋯+(n−1s−2r+1). If either K is a set of r consecutive integers or L={1,2,…,s}, then|F|⩽(n−1s)+(n−1s−1)+⋯+(n−1s−r). We will also prove similar results which involve two families of subsets of [n]. These results improve the existing upper bounds substantially
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