Abstract
For given positive integers k and n, a family F of subsets of {1, . . . , n} is kantichain saturated if it does not contain an antichain of size k, but adding any set to F creates an antichain of size k. We use sat * (n, k) to denote the smallest size of such a family. For all k and sufficiently large n, we determine the exact value of sat * (n, k). Our result implies that sat * (n, k) = n(k -1) -Θ(k log k), which confirms several conjectures on antichain saturation. Previously, exact values for sat * (n, k) were only known for k up to 6.</p><p>We also prove a strengthening of a result of Lehman-Ron which may be of independent interest. We show that given m disjoint chains C 1 , . . . , C m in the Boolean lattice, we can create m disjoint skipless chains that cover the elements from ∪ m i=1 C i (where we call a chain skipless if any two consecutive elements differ in size by exactly one).