$s$-almost cross-$t$-intersecting families for vector spaces
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912899293773824 |
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| author | Liu, Dehai Wang, Jinhua Yao, Tian |
| author_facet | Liu, Dehai Wang, Jinhua Yao, Tian |
| contents | Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F} _{q} $, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F},\mathcal{G}\subseteq {V\brack k}$ are said to be cross-$t$-intersecting if $\dim(F\cap G)\ge t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$. Two families $\mathcal{F}$ and $\mathcal{G}$ are called $s$-almost cross-$t$-intersecting if each member of $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members of $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we discribe the structure of $s$-almost cross-$t$-intersecting families with maximum product of their sizes. In addition, we prove a stability result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11532 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $s$-almost cross-$t$-intersecting families for vector spaces Liu, Dehai Wang, Jinhua Yao, Tian Combinatorics Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F} _{q} $, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F},\mathcal{G}\subseteq {V\brack k}$ are said to be cross-$t$-intersecting if $\dim(F\cap G)\ge t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$. Two families $\mathcal{F}$ and $\mathcal{G}$ are called $s$-almost cross-$t$-intersecting if each member of $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members of $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we discribe the structure of $s$-almost cross-$t$-intersecting families with maximum product of their sizes. In addition, we prove a stability result. |
| title | $s$-almost cross-$t$-intersecting families for vector spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.11532 |