An existence theorem for antichains

AbstractLet k1, k2,…, kn be given integers, 1 ⩽ k1 ⩽ k2 ⩽ … ⩽ kn, and let S be the set of vectors x = (x1,…, xn) with integral coefficients satisfying 0 ⩽ xi ⩽ ki, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities xi ⩽ yi, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k1 + k2 + … + kn and for 0 ⩽ l ⩽ K let (l)H denote the subset of H consisting of vectors h = (h1, h2,…, hn) which satisfy h1 + h2 + … + hn = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if l < K2, |(K2)H′| = |(K2)H| if K is even and |(l)H′| = |(l)H| + |(K − l)H| if l>K2