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 = {l_{1}, l_{2},..., l_{s}} and K = {k_{1}, k_{2},... k_{r}} be two subsets of {0, 1, 2,..., p-1} satisfying max l_{j} < min k_{i}. We will prove the following results: If F = {F_{1}, F_{2},..., F_{m}} is a family of subsets of [n]={1,2,..., n} such that |F_{i}∩F_{j}| (mod p) ∈ L for every pair i≠j and |F_{i}| (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. Download: pdf |