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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.08651 |
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| _version_ | 1866908806433210368 |
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| author | Liu, Dingyuan |
| author_facet | Liu, Dingyuan |
| contents | Given $s,t\in\mathbb{N}$, a complete bipartite poset $\mathcal{K}_{s,t}$ is a poset whose Hasse diagram consists of $s$ pairwise incomparable vertices in the upper layer and $t$ pairwise incomparable vertices in the lower layer, such that every vertex in the upper layer is larger than all vertices in the lower layer. A family $\mathcal{F}\subseteq2^{[n]}$ is called induced $\mathcal{K}_{s,t}$-saturated if $(\mathcal{F},\subseteq)$ contains no induced copy of $\mathcal{K}_{s,t}$, whereas adding any set from $2^{[n]}\backslash\mathcal{F}$ to $\mathcal{F}$ creates an induced $\mathcal{K}_{s,t}$. Let $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ denote the smallest size of an induced $\mathcal{K}_{s,t}$-saturated family $\mathcal{F}\subseteq2^{[n]}$. It was conjectured that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ is superlinear in $n$ for certain values of $s$ and $t$. In this paper, we show that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})=O(n)$ for all fixed $s,t\in\mathbb{N}$. Moreover, we prove a linear lower bound on $\mathrm{sat}^{*}(n,\mathcal{P})$ for a large class of posets $\mathcal{P}$, particularly for $\mathcal{K}_{s,2}$ with $s\in\mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Induced saturation for complete bipartite posets Liu, Dingyuan Combinatorics Given $s,t\in\mathbb{N}$, a complete bipartite poset $\mathcal{K}_{s,t}$ is a poset whose Hasse diagram consists of $s$ pairwise incomparable vertices in the upper layer and $t$ pairwise incomparable vertices in the lower layer, such that every vertex in the upper layer is larger than all vertices in the lower layer. A family $\mathcal{F}\subseteq2^{[n]}$ is called induced $\mathcal{K}_{s,t}$-saturated if $(\mathcal{F},\subseteq)$ contains no induced copy of $\mathcal{K}_{s,t}$, whereas adding any set from $2^{[n]}\backslash\mathcal{F}$ to $\mathcal{F}$ creates an induced $\mathcal{K}_{s,t}$. Let $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ denote the smallest size of an induced $\mathcal{K}_{s,t}$-saturated family $\mathcal{F}\subseteq2^{[n]}$. It was conjectured that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ is superlinear in $n$ for certain values of $s$ and $t$. In this paper, we show that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})=O(n)$ for all fixed $s,t\in\mathbb{N}$. Moreover, we prove a linear lower bound on $\mathrm{sat}^{*}(n,\mathcal{P})$ for a large class of posets $\mathcal{P}$, particularly for $\mathcal{K}_{s,2}$ with $s\in\mathbb{N}$. |
| title | Induced saturation for complete bipartite posets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.08651 |