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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2408.14648 |
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| _version_ | 1866910975389597696 |
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| author | Martin, Ryan R Veldt, Nick |
| author_facet | Martin, Ryan R Veldt, Nick |
| contents | Given a set $X$, the power set $\mathbb{P}(X)$, and a finite poset $P$, a family $F\subset \mathbb{P}(X)$ is said to be induced-$P$-free if there is no injection $ϕ: P\rightarrow \mathbb{F}$ such that $ϕ(p)\subseteqϕ(q)$ if and only if $p\leq_{P} q$, for all $p, q \in P$. The family $F$ is induced-$P$-saturated if it is maximal with respect to being induced-$P$-free. If $n=|X|$, then the size of the smallest induced-$P$-saturated family in $\mathbb{P}(X)$ is denoted $sat(n,P)$.
The poset $2C_2$ is two incomparable 2-chains (the Hasse diagram is two vertex-disjoint edges) and Keszegh, Lemons, Martin, Pálvölgyi, and Patkós proved that $n+2\leq sat(n,2C_2)\leq 2n$ and gave one isomorphism class of an induced-$2C_2$-saturated family that achieves the upper bound.
We show that the lower bound can be improved to $3n/2 + 1/2$ by examining the necessary structure of a saturated family. In addition, we provide many examples of induced-$2C_2$-saturated families of size $2n$ in $\mathbb{P}(X)$ where $|X|=n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14648 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Induced Saturation of the Poset 2C_2 Martin, Ryan R Veldt, Nick Combinatorics Given a set $X$, the power set $\mathbb{P}(X)$, and a finite poset $P$, a family $F\subset \mathbb{P}(X)$ is said to be induced-$P$-free if there is no injection $ϕ: P\rightarrow \mathbb{F}$ such that $ϕ(p)\subseteqϕ(q)$ if and only if $p\leq_{P} q$, for all $p, q \in P$. The family $F$ is induced-$P$-saturated if it is maximal with respect to being induced-$P$-free. If $n=|X|$, then the size of the smallest induced-$P$-saturated family in $\mathbb{P}(X)$ is denoted $sat(n,P)$. The poset $2C_2$ is two incomparable 2-chains (the Hasse diagram is two vertex-disjoint edges) and Keszegh, Lemons, Martin, Pálvölgyi, and Patkós proved that $n+2\leq sat(n,2C_2)\leq 2n$ and gave one isomorphism class of an induced-$2C_2$-saturated family that achieves the upper bound. We show that the lower bound can be improved to $3n/2 + 1/2$ by examining the necessary structure of a saturated family. In addition, we provide many examples of induced-$2C_2$-saturated families of size $2n$ in $\mathbb{P}(X)$ where $|X|=n$. |
| title | Induced Saturation of the Poset 2C_2 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.14648 |