An Asymptotic Complete Intersection Theorem for Chain Products

For positive integers k, ` and n let N`(n, k) = {a = (a1, . . . , an) : ai ∈ {0, 1, . . . , k}, i = 1, . . . , n,∑ ai = `}. A family F ⊆ N`(n, k) is called t-intersecting if for all a, b ∈ F there exist t coordinates i1, . . . , it such that ai j , bi j ≥ 1 holds for j = 1, . . . , t . Define M`(n, k, t) = max{|F | : F ⊆ N`(n, k),F is t-intersecting}. N`(n, k) can be viewed as the `-th level of the direct product of n chains 0 l 1 l · · · l k (see [4] for terminology not explained here). We identify N`(n, 1) with ([n] ` ) , the family of all `-subsets of {1, . . . , n}. Set W`(n, k) = |N`(n, k)|. Define for a ∈ N`(n, k) resp. F ⊆ N`(n, k) the support of a resp. of F by supp(a) = {i : ai > 0} resp. supp(F) = {supp(a) : a ∈ F}. Obviously, F ⊆ N`(n, k) is t-intersecting iff supp(F) is t-intersecting. Let Sr,` = {S ∈ ([n] ` ) : |S ∩ [1, t + 2r ]| ≥ t + r}, where r ∈ {0} ∪N and [i, j] is defined as {i, i + 1, . . . , j}. THEOREM 1.1 (AHLSWEDE, KHACHATRIAN [1]). Let n > 2`− t and r ∈ {0} ∪ N such that (`− t + 1)(2+ t−1 r+1 ) ≤ n < (`− t + 1)(2+ t−1 r ). Then M`(n, 1, t) = |Sr,`|.