Bounded fractional intersecting families are linear in size
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_ | 1866911156778565632 |
|---|---|
| author | Balachandran, Niranjan Das, Shagnik Sankarnarayanan, Brahadeesh |
| author_facet | Balachandran, Niranjan Das, Shagnik Sankarnarayanan, Brahadeesh |
| contents | Using the sunflower method, we show that if $θ\in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $θ$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq (\frac{3}{2} + o(1))n$. This partially solves a conjecture of Balachandran, Mathew and Mishra that any $θ$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounded fractional intersecting families are linear in size Balachandran, Niranjan Das, Shagnik Sankarnarayanan, Brahadeesh Combinatorics 05D05 (Primary) 03E05 (Secondary) Using the sunflower method, we show that if $θ\in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $θ$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq (\frac{3}{2} + o(1))n$. This partially solves a conjecture of Balachandran, Mathew and Mishra that any $θ$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets. |
| title | Bounded fractional intersecting families are linear in size |
| topic | Combinatorics 05D05 (Primary) 03E05 (Secondary) |
| url | https://arxiv.org/abs/2402.14981 |