Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron

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.

We also prove a strengthening of a result of Lehman–Ron which may be of independent interest. We show that given m disjoint chains C¹, . . . , C^m in the Boolean lattice, we can create m disjoint skipless chains that cover the elements from ∪^m_i=1Cⁱ (where we call a chain
skipless if any two consecutive elements differ in size by exactly one).