Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.14028 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917863293452288 |
|---|---|
| author | Frankl, Peter Wang, Jian |
| author_facet | Frankl, Peter Wang, Jian |
| contents | Let $\mathcal{F}\subset \binom{X}{k}$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\mathcal{F}$ is intersecting, i.e., $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. Let $Δ(\mathcal{F})$ be the maximum degree of $\mathcal{F}$. For a constant $C\geq 1$ the $C$-diversity, $γ_C(\mathcal{F})$ is defined as $|\mathcal{F}|-CΔ(\mathcal{F})$. Define $\mathcal{F}_{123} =\left\{F\in \binom{X}{k}\colon |F\cap \{1,2,3\}|=2\right\}$. It has $C$-diversity $(3-2C)\binom{n-3}{k-2}$. The main result shows that for $1< C<\frac{3}{2}$ and $n\geq \frac{42}{3-2C}k$, $γ_C(\mathcal{F})\leq γ_C(\mathcal{F}_{123})$ with equality if and only if $\mathcal{F}$ is isomorphic to $\mathcal{F}_{123}$. For the case of ordinary diversity $(C=1)$ a strong stability is proven. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_14028 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $C$-diversity of intersecting hypergraphs Frankl, Peter Wang, Jian Combinatorics Let $\mathcal{F}\subset \binom{X}{k}$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\mathcal{F}$ is intersecting, i.e., $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. Let $Δ(\mathcal{F})$ be the maximum degree of $\mathcal{F}$. For a constant $C\geq 1$ the $C$-diversity, $γ_C(\mathcal{F})$ is defined as $|\mathcal{F}|-CΔ(\mathcal{F})$. Define $\mathcal{F}_{123} =\left\{F\in \binom{X}{k}\colon |F\cap \{1,2,3\}|=2\right\}$. It has $C$-diversity $(3-2C)\binom{n-3}{k-2}$. The main result shows that for $1< C<\frac{3}{2}$ and $n\geq \frac{42}{3-2C}k$, $γ_C(\mathcal{F})\leq γ_C(\mathcal{F}_{123})$ with equality if and only if $\mathcal{F}$ is isomorphic to $\mathcal{F}_{123}$. For the case of ordinary diversity $(C=1)$ a strong stability is proven. |
| title | On the $C$-diversity of intersecting hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2308.14028 |