$s$-almost $t$-intersecting families for vector spaces

Abstract

Let $\mathcal{F}$ be a family of $k$-dimensional subspaces of an $n$-dimensional vector space. Write ${\mathcal{D}}_{\mathcal{F}}(H;t)=\{F\in \mathcal{F}:\dim (F\cap H)\le t\}$ for a subspace $H$. The family $\mathcal{F}$ is called $s$-almost $t$-intersecting if $|{\mathcal{D}}_{\mathcal{F}}(F;t)|\le s$ for each $F\in \mathcal{F}$. In this note, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting.

This manuscript has been improved joint with Dehai Liu, Kaishun Wang and Tian Yao; see arXiv:2406.05840v3