For a given fixed poset P we say that a family of subsets of [n] is P-saturated if it does not contain an induced copy of P, but whenever we add to it a new set, an induced copy of P is formed. The size of the smallest such family is denoted by sat∗(n, P). For the diamond poset D2 (the two-dimensional Boolean lattice), Martin, Smith and Walker proved that ≤ sat∗(n, D2) ≤ n + 1. In this paper we prove that sat∗(n, D2) ≥ (−o(1)) . We also explore the properties that a diamond-saturated family of… Expand

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… Expand

. For a ﬁxed poset P , a family F of subsets of [ n ] is induced P -saturated if F does not contain an induced copy of P , but for every subset S of [ n ] such that S 6∈ F , then P is an induced… Expand

For the 4-point poset known as the diamond, this work proves the minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\operatorname{sat}^*(n,\ mathcal{D}_2)\geq\sqrt{n}$, improving upon a logarithmic lower bound.Expand

This paper addresses the problem of finding the minimum size of a family saturating the k-Sperner property and the minimumsized family that saturates the Spernerproperty and that consists only of l-sets and (l + 1)-sets.Expand

The lower bound is shown that, if $|X|\geq k$ is sufficiently large with respect to $k, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements.Expand