The Saturation Number for the Diamond is Linear
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917323768594432 |
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| author | Ivan, Maria-Romina Jaffe, Sean |
| author_facet | Ivan, Maria-Romina Jaffe, Sean |
| contents | For a fixed poset $\mathcal P$ we say that a family $\mathcal F\subseteq\mathcal P([n])$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add a new set to $\mathcal F$, we form an induced copy of $\mathcal P$. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$.\par For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), while it is easy to see that the saturation number is at most $n+1$, the best known lower bound has stayed at $O(\sqrt n)$ since the introduction of the area of poset saturation. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq \frac{n+1}{5}$, establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Saturation Number for the Diamond is Linear Ivan, Maria-Romina Jaffe, Sean Combinatorics 06A07, 05D05 For a fixed poset $\mathcal P$ we say that a family $\mathcal F\subseteq\mathcal P([n])$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add a new set to $\mathcal F$, we form an induced copy of $\mathcal P$. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$.\par For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), while it is easy to see that the saturation number is at most $n+1$, the best known lower bound has stayed at $O(\sqrt n)$ since the introduction of the area of poset saturation. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq \frac{n+1}{5}$, establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems. |
| title | The Saturation Number for the Diamond is Linear |
| topic | Combinatorics 06A07, 05D05 |
| url | https://arxiv.org/abs/2507.05122 |