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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), 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 in the Boolean lattice, we can create $m$ disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one).
[Paul Bastide] (Université de Bordeaux )
http://perso.eleves.ens-rennes.fr/people/paul.bastide/
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