Poset saturation of unions of chains
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911245234339840 |
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| author | Ji, Shengjin Patkós, Balázs Yue, Erfei |
| author_facet | Ji, Shengjin Patkós, Balázs Yue, Erfei |
| contents | A family $\mathcal{G}$ of sets is a(n induced) copy of a poset $P=(P,\leqslant)$ if there exists a bijection $b:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ holds if and only if $b(p)\subseteq b(q)$. The induced saturation number sat$^*(n,P)$ is the minimum size of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain any copy of $P$, but for any $G\in 2^{[n]}\setminus \mathcal{F}$, the family $\mathcal{F}\cup \{G\}$ contains a copy of $P$. We consider sat$^*(n,P)$ for posets $P$ that are formed by pairwise incomparable chains, i.e. $P=\bigoplus_{j=1}^mC_{i_j}$. We make the following two conjectures: (i) sat$^*(n,P)=O(n)$ for all such posets and (ii) sat$^*(n,P)=O(1)$ if not all chains are of the same size. (The second conjecture is known to hold if there is a unique longest among the chains.) We verify these conjectures in some special cases: we prove (i) if all chains are of the same length, we prove (ii) in the first unknown general case: for posets $2C_k+C_1$. Finally, we give an infinite number of examples showing that (ii) is not a necessary condition for sat$^*(n,P)=O(1)$ among posets $P=\bigoplus_{j=1}^mC_{i_j}$: we prove sat$^*(n,(\binom{2t}{t}+1)C_2)=O(1)$ for all $t\ge 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23128 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poset saturation of unions of chains Ji, Shengjin Patkós, Balázs Yue, Erfei Combinatorics A family $\mathcal{G}$ of sets is a(n induced) copy of a poset $P=(P,\leqslant)$ if there exists a bijection $b:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ holds if and only if $b(p)\subseteq b(q)$. The induced saturation number sat$^*(n,P)$ is the minimum size of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain any copy of $P$, but for any $G\in 2^{[n]}\setminus \mathcal{F}$, the family $\mathcal{F}\cup \{G\}$ contains a copy of $P$. We consider sat$^*(n,P)$ for posets $P$ that are formed by pairwise incomparable chains, i.e. $P=\bigoplus_{j=1}^mC_{i_j}$. We make the following two conjectures: (i) sat$^*(n,P)=O(n)$ for all such posets and (ii) sat$^*(n,P)=O(1)$ if not all chains are of the same size. (The second conjecture is known to hold if there is a unique longest among the chains.) We verify these conjectures in some special cases: we prove (i) if all chains are of the same length, we prove (ii) in the first unknown general case: for posets $2C_k+C_1$. Finally, we give an infinite number of examples showing that (ii) is not a necessary condition for sat$^*(n,P)=O(1)$ among posets $P=\bigoplus_{j=1}^mC_{i_j}$: we prove sat$^*(n,(\binom{2t}{t}+1)C_2)=O(1)$ for all $t\ge 1$. |
| title | Poset saturation of unions of chains |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.23128 |