Structure of $t$-Intersecting Families of Vector Spaces
Ferdinand Ihringer, Andrey Kupavskii

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 -intersecting and -cross-intersecting families of -dimensional subspaces in finite vector spaces of dimension . We show that all large -intersecting families admit a governing low-dimensional structure for . 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 -intersecting and -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.
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