Some new Bollobás-type inequalities
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914851997089792 |
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| author | Yue, Erfei |
| author_facet | Yue, Erfei |
| contents | A family of disjoint pairs of finite sets $\mathcal{P}=\{(A_i,B_i)\mid i\in[m]\}$ is called a Bollobás system if $A_i\cap B_j\neq\emptyset$ for every $i\neq j$, and a skew Bollobás system if $A_i\cap B_j\neq\emptyset$ for every $i<j$. Bollobás proved that for a Bollobás system, the inequality \begin{equation*} \sum_{i=1}^m\binom{|A_i|+|B_i|}{|A_i|}^{-1}\leq 1 \end{equation*} holds. Hegedüs and Frankl generalized this theorem to skew Bollobás systems with the inequality \begin{equation*} \sum_{i=1}^m\binom{|A_i|+|B_i|}{|A_i|}^{-1}\leq 1+n, \end{equation*} provided $A_i,B_i\subseteq [n]$. In this paper, we improve this inequality to \begin{equation*} \sum_{i=1}^m \left((1+|A_i|+|B_i|) \binom{|A_i|+|B_i|}{|A_i|}\right)^{-1} \leq 1 \end{equation*} with probabilistic method. We also generalize this result to partitions of sets on both symmetric and skew cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some new Bollobás-type inequalities Yue, Erfei Combinatorics A family of disjoint pairs of finite sets $\mathcal{P}=\{(A_i,B_i)\mid i\in[m]\}$ is called a Bollobás system if $A_i\cap B_j\neq\emptyset$ for every $i\neq j$, and a skew Bollobás system if $A_i\cap B_j\neq\emptyset$ for every $i<j$. Bollobás proved that for a Bollobás system, the inequality \begin{equation*} \sum_{i=1}^m\binom{|A_i|+|B_i|}{|A_i|}^{-1}\leq 1 \end{equation*} holds. Hegedüs and Frankl generalized this theorem to skew Bollobás systems with the inequality \begin{equation*} \sum_{i=1}^m\binom{|A_i|+|B_i|}{|A_i|}^{-1}\leq 1+n, \end{equation*} provided $A_i,B_i\subseteq [n]$. In this paper, we improve this inequality to \begin{equation*} \sum_{i=1}^m \left((1+|A_i|+|B_i|) \binom{|A_i|+|B_i|}{|A_i|}\right)^{-1} \leq 1 \end{equation*} with probabilistic method. We also generalize this result to partitions of sets on both symmetric and skew cases. |
| title | Some new Bollobás-type inequalities |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.17639 |