On Saturated $k$-Sperner Systems

Abstract

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. [11] 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 2 k−1 . We disprove this conjecture by showing that there exists ε > 0 such that for every k and |X| ≥ n 0 (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. [11] proved that, if |X| ≥ k, then the smallest such collection contains between 2 k/2−1 and O log k k 2 k elements. We show that if |X| ≥ k 2 + k, then the lower bound is best possible, up to a polynomial factor.

The largest projective cube-free subsets of Z2n

“…Another related problem, following [8,15], is to determine the smallest maximal kcube free set in Z 2 n . That is, the smallest S ⊂ Z 2 n that is k-cube free, but the additon of any new element to S makes it not k-cube-free.…”

Saturation in the Hypercube and Bootstrap Percolation

“…Johnson and Pinto [27] also study another type of saturation problem, which is often referred to as semi-saturation [18,22] (other terms have also been used: see [10,28,32,34,36]). Given graphs F and H, say that a spanning subgraph G of F is (F, H)-semi-saturated if, for every edge e ∈ E(F) \ E(G), the graph G + e contains more copies of H than G does.…”

Saturated Subgraphs of the Hypercube

“…A family A ⊆ P(X) is said to be F-saturated if there is no subfamily of A with the same poset structure as F, but adding any set to A destroys this property. Both the maximum and minimum size of such A have been studied: see for instance Katona and Tarján [8] for the former and Morrison, Noel and Scott [10] for the latter.…”