Set Systems with L-intersections modulo a Prime Number

William Y. C. Chen and Jiuqiang Liu

  Abstract:   Let 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 max lj < min ki. 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| (mod p) ∈ L for every pair i≠j and |Fi| (mod p) ∈ K for every 1≤i≤m, then

If either K is a set of r consecutive integers or L={1,2,..., s}, then

We will also prove similar results which involve two families of subsets of [n]. These results improve the existing upper bounds substantially.

  AMS Classification:   05A15, 05A18.

  Keywords:   Erdös-Ko-Rado Theorem, Frankl-Ray-Chaudhuri-Wilson Theorems, Frankl-Füredi’s conjecture, Snevily's conjecture.

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