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

TL;DR

This paper investigates families of vector space subspaces where most pairs intersect in at least a certain dimension, proving that maximum-sized families with a relaxed intersection condition are actually fully intersecting.

Contribution

It establishes that the largest $s$-almost $t$-intersecting families are necessarily $t$-intersecting, extending classical intersection results to near-intersecting families.

Findings

Maximum size $s$-almost $t$-intersecting families are $t$-intersecting.

Provides a characterization of near-intersecting families in vector spaces.

Extends classical intersection theorems to approximate intersection conditions.

Abstract

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