On maximal antichains containing no set and its complement

AbstractLet 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 xϵS 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 |(12K)X| does not exceed the number of vectors in (12K)S with first coordinate different from k1, then ∑i=0Ki≠12K|(i)X||(i)S|+|(12K)X||(12K-1)S|⩽1