Minimal Diamond-Saturated Families

  title={Minimal Diamond-Saturated Families},
  author={Maria-Romina Ivan},
  journal={Contemporary Mathematics},
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… 

Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron

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

The induced saturation problem for posets

. 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



Improved Bounds for Induced Poset Saturation

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.

Extremal Finite Set Theory

The saturation number of induced subposets of the Boolean lattice

Saturating Sperner Families

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.

On Saturated k-Sperner Systems

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.