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 \ (k\) -antichain 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) - (k 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, , Cᵐ\) in the Boolean lattice, we can create \ (m\) disjoint skipless chains that cover the elements from \ (₈=₁ᵐCⁱ\) (where we call a chain skipless if any two consecutive elements differ in size by exactly one). Mathematics Subject Classifications: 06A07, 05D99Keywords: Skipless chains, poset saturation, antichain saturation, Boolean lattice