On maximal antichains containing no set and its complement

Abstract

Let 1≤k1≤k2≤...≤kn be integers and let S denote the set of all vectors x = (x1, ..., xn with integral coordinates satisfying 0≤xi≤ki, i = 1,2, ..., n; equivalently, S is the set of all subsets of a multiset consisting of ki elements of type i, i = 1,2, ..., n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities xi≤yi, i = 1,2, ..., n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, ..., K, where K = k1 + k2 + ...kn. |H| denotes the number of vectors in H, and the complement of a vector xε{lunate}S is (k1-x1, k2-x2, ..., kn -xn). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and |( 1 2K)X| does not exceed the number of vectors in ( 1 2K)S with first coordinate different from k1, then ∑ i=0 K i≠ 1 2K |(i)X| |(i)S|+ |( 1 2K)X| |( 1 2K-1)S|≤1. © 1981.