Minimal Diamond-Saturated Families
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913764514725888 |
|---|---|
| author | Ivan, Maria-Romina |
| author_facet | Ivan, Maria-Romina |
| contents | For a given fixed poset $\mathcal P$ we say that a family of subsets of $[n]$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add to it a new set, an induced copy of $\mathcal P$ is formed. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$. For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved that $\sqrt n\leq\text{sat}^*(n, \mathcal D_2)\leq n+1$. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq (4-o(1))\sqrt n$. We also explore the properties that a diamond-saturated family of size $c\sqrt n$, for a constant $c$, would have to have. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_01118 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Minimal Diamond-Saturated Families Ivan, Maria-Romina Combinatorics 06A07, 05D05 For a given fixed poset $\mathcal P$ we say that a family of subsets of $[n]$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add to it a new set, an induced copy of $\mathcal P$ is formed. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$. For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved that $\sqrt n\leq\text{sat}^*(n, \mathcal D_2)\leq n+1$. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq (4-o(1))\sqrt n$. We also explore the properties that a diamond-saturated family of size $c\sqrt n$, for a constant $c$, would have to have. |
| title | Minimal Diamond-Saturated Families |
| topic | Combinatorics 06A07, 05D05 |
| url | https://arxiv.org/abs/2110.01118 |