Saturating Sperner families
  A family $\cF \subseteq 2^{[n]}$ saturates the monotone decreasing property $\cP$ if $\cF$ satisfies $\cP$ and one cannot add any set to $\cF$ such that property $\cP$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the $k$-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of $l$-sets and $(l+1)$-sets. 
D\'aniel Gerbner, Bal\'azs Keszegh, Nathan Lemons, D\"om\"ot\"or   P\'alv\"olgyi, Cory Palmer, Bal\'azs Patk\'os
math.CO