Given a set X, a collection F⊆P(X) is said to
be k-Sperner if it does not contain a chain of length k+1 under set
inclusion and it is saturated if it is maximal with respect to this property.
Gerbner et al. conjectured that, if ∣X∣ is sufficiently large with respect to
k, then the minimum size of a saturated k-Sperner system
F⊆P(X) is 2k−1. We disprove this conjecture
by showing that there exists ε>0 such that for every k and ∣X∣≥n0(k) there exists a saturated k-Sperner system
F⊆P(X) with cardinality at most
2(1−ε)k.
A collection F⊆P(X) is said to be an
oversaturated k-Sperner system if, for every
S∈P(X)∖F, F∪{S} contains more
chains of length k+1 than F. Gerbner et al. proved that, if
∣X∣≥k, then the smallest such collection contains between 2k/2−1 and
O(klogk2k) elements. We show that if ∣X∣≥k2+k,
then the lower bound is best possible, up to a polynomial factor.Comment: 17 page