# Minimal Diamond-Saturated Families

@article{Ivan2022MinimalDF,
title={Minimal Diamond-Saturated Families},
author={Maria-Romina Ivan},
journal={Contemporary Mathematics},
year={2022}
}
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…
2 Citations
Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron
• Mathematics
• 2022
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
• Mathematics
• 2022
. 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

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