The Induced Saturation Number for $\mathcal{V}_3$ is Linear
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911065873317888 |
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| author | Brownlie, James Jaffe, Sean |
| author_facet | Brownlie, James Jaffe, Sean |
| contents | Given a poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is said to be $\mathcal{P}$-saturated if $\mathcal{F}$ does not contain an induced copy $\mathcal P$, but every proper superset of $\mathcal{F}$ contains one. The minimum size of a $\mathcal P$-saturated family in the $n$-dimensional Boolean lattice is denoted by $sat^*(n,\mathcal{P})$.\par In this paper, we consider the poset $\mathcal V_3$ (the four element poset with one minimal element and three incomparable maximal elements) and show that $sat^*(n,\mathcal{V}_3)\geq \frac{n}{2}$. This represents the first linear lower bound for $sat^*(n,\mathcal{V}_3)$, improving upon the previously best-known bound of $2\sqrt{n}$. Our result establishes that $sat^*(n,\mathcal{V}_3) = Θ(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Induced Saturation Number for $\mathcal{V}_3$ is Linear Brownlie, James Jaffe, Sean Combinatorics 06A07, 05D05 Given a poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is said to be $\mathcal{P}$-saturated if $\mathcal{F}$ does not contain an induced copy $\mathcal P$, but every proper superset of $\mathcal{F}$ contains one. The minimum size of a $\mathcal P$-saturated family in the $n$-dimensional Boolean lattice is denoted by $sat^*(n,\mathcal{P})$.\par In this paper, we consider the poset $\mathcal V_3$ (the four element poset with one minimal element and three incomparable maximal elements) and show that $sat^*(n,\mathcal{V}_3)\geq \frac{n}{2}$. This represents the first linear lower bound for $sat^*(n,\mathcal{V}_3)$, improving upon the previously best-known bound of $2\sqrt{n}$. Our result establishes that $sat^*(n,\mathcal{V}_3) = Θ(n)$. |
| title | The Induced Saturation Number for $\mathcal{V}_3$ is Linear |
| topic | Combinatorics 06A07, 05D05 |
| url | https://arxiv.org/abs/2507.08353 |