Structure of $t$-Intersecting Families of Vector Spaces

TL;DR

This paper investigates the structure of large $t$-intersecting families of vector spaces, establishing their low-dimensional governing structure and extending classical extremal set theory results to vector spaces.

Contribution

It introduces a low-dimensional structure theorem for large $t$-intersecting families and generalizes classical extremal results like the Hilton-Milner theorem to vector spaces.

Findings

Large $t$-intersecting families have a low-dimensional governing structure.

Determined the families with the largest diversity among $t$-intersecting families.

Proved a Frankl-type degree-diversity result generalizing Hilton-Milner theorem.

Abstract

We study $t$-intersecting and $t$-cross-intersecting families of $k$-dimensional subspaces in finite vector spaces of dimension $n$. We show that all large $t$-intersecting families admit a governing low-dimensional structure for $n \ge 2k+1$. This result, together with its cross-intersecting variant, allows us to prove analogues of several classical extremal set-theoretic results. In particular, we determine the intersecting families with the largest diversity, and we establish a Frankl-type degree-diversity result that generalizes the Hilton-Milner theorem. Our proofs rely on simplification procedures for $t$-intersecting and $t$-cross-intersecting families of subspaces. These procedures are based on the concept of subspace spreadness, a generalization of the classical notion of spreadness for set systems.