$s$-almost $t$-intersecting families for vector spaces
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_ | 1866915068000600064 |
|---|---|
| author | Ji, Lijun Liu, Dehai Wang, Kaishun Yao, Tian Yu, Shuhui |
| author_facet | Ji, Lijun Liu, Dehai Wang, Kaishun Yao, Tian Yu, Shuhui |
| contents | Let $V$ be a finite dimensional vector space over a finite field, and $\mathcal{F}$ a family consisting of $k$-subspaces of $V$. The family $\mathcal{F}$ is called $t$-intersecting if $\dim(F_{1}\cap F_{2})\geq t$ for any $F_{1}, F_{2}\in \mathcal{F}$. We say $\mathcal{F}$ is $s$-almost $t$-intersecting if for each $F\in \mathcal{F}$ there are at most $s$ members $F^{\prime}$ of $\mathcal{F}$ such that $\dim(F\cap F^{\prime})<t$. In this paper, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting. We also consider $s$-almost $t$-intersecting families which are not $t$-intersecting, and characterize such families with maximum size for $(s,t)\neq(1,1)$. The result for $1$-almost $1$-intersecting families provided by Shan and Zhou is generalized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05840 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $s$-almost $t$-intersecting families for vector spaces Ji, Lijun Liu, Dehai Wang, Kaishun Yao, Tian Yu, Shuhui Combinatorics 05D05, 05A30 Let $V$ be a finite dimensional vector space over a finite field, and $\mathcal{F}$ a family consisting of $k$-subspaces of $V$. The family $\mathcal{F}$ is called $t$-intersecting if $\dim(F_{1}\cap F_{2})\geq t$ for any $F_{1}, F_{2}\in \mathcal{F}$. We say $\mathcal{F}$ is $s$-almost $t$-intersecting if for each $F\in \mathcal{F}$ there are at most $s$ members $F^{\prime}$ of $\mathcal{F}$ such that $\dim(F\cap F^{\prime})<t$. In this paper, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting. We also consider $s$-almost $t$-intersecting families which are not $t$-intersecting, and characterize such families with maximum size for $(s,t)\neq(1,1)$. The result for $1$-almost $1$-intersecting families provided by Shan and Zhou is generalized. |
| title | $s$-almost $t$-intersecting families for vector spaces |
| topic | Combinatorics 05D05, 05A30 |
| url | https://arxiv.org/abs/2406.05840 |