Minimal Diamond-Saturated Families

Authors

DOI:

https://doi.org/10.37256/cm.3220221333

Keywords:

poset saturation, diamond, extremal combinatorics

Abstract

For a given fixed poset 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 CM-133328.jpg ≤ sat(n, D2) ≤ n + 1. In this paper we prove that sat(n, D2) ≥ (CM-133311.jpgo(1)) CM-133329.jpg. We also explore the properties that a diamond-saturated family of size c CM-1333210.jpg, for a constant c, would have to have.

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Published

2022-03-31

How to Cite

1.
Ivan M-R. Minimal Diamond-Saturated Families. Contemporary Mathematics [Internet]. 2022 Mar. 31 [cited 2022 Jun. 28];3(2):81-8. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/1333