On maximal antichains consisting of sets and their complements

AbstractLet 1 ⩽ k1 ⩽ k2 ⩽ … ⩽ kn be integers and let S denote the set of all vectors x = (x1, x2, …, xn) with integral coordinates satisfying 0 ⩽ xi ⩽ ki, i = 1, 2, …, n. The complement of x is (k1 − x1, k2 − x2, …, kn − xn) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities xi ⩽ yi, i = 1, 2, …, n, do not all hold. We determine an LYM inequality and the maximal cardinality of an antichain consisting of vectors and its complements. Also a generalization of the Erdös-Ko-Rado theorem is given