The maximum sturdiness of intersecting families
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915055800418304 |
|---|---|
| author | Frankl, Peter Wang, Jian |
| author_facet | Frankl, Peter Wang, Jian |
| contents | Given a family $\mathcal{F}\subset 2^{[n]}$ and $1\leq i\neq j\leq n$, we use $\mathcal{F}(\bar{i},j)$ to denote the family $\{F\setminus \{j\}\colon F\in \mathcal{F},\ F\cap \{i,j\}=\{j\}\}$. The sturdiness of $\mathcal{F}$ is defined as the minimum $|\mathcal{F}(\bar{i},j)|$ over all $i,j\in [n]$ with $i\neq j$. It has a very natural algebraic definition as well. In the present paper, we consider the maximum sturdiness of $k$-uniform intersecting families, $k$-uniform $t$-intersecting families and non-uniform $t$-intersecting families. One of the main results shows that for $n\geq 36(k+6)$, an intersecting family $\mathcal{F}\subset \binom{[n]}{k}$ has sturdiness at most $\binom{n-4}{k-3}$, which is best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07090 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The maximum sturdiness of intersecting families Frankl, Peter Wang, Jian Combinatorics Given a family $\mathcal{F}\subset 2^{[n]}$ and $1\leq i\neq j\leq n$, we use $\mathcal{F}(\bar{i},j)$ to denote the family $\{F\setminus \{j\}\colon F\in \mathcal{F},\ F\cap \{i,j\}=\{j\}\}$. The sturdiness of $\mathcal{F}$ is defined as the minimum $|\mathcal{F}(\bar{i},j)|$ over all $i,j\in [n]$ with $i\neq j$. It has a very natural algebraic definition as well. In the present paper, we consider the maximum sturdiness of $k$-uniform intersecting families, $k$-uniform $t$-intersecting families and non-uniform $t$-intersecting families. One of the main results shows that for $n\geq 36(k+6)$, an intersecting family $\mathcal{F}\subset \binom{[n]}{k}$ has sturdiness at most $\binom{n-4}{k-3}$, which is best possible. |
| title | The maximum sturdiness of intersecting families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.07090 |