Kleitman's theorem over vector spaces: parity phenomena in canonical and global stability

TL;DR

This paper completely solves the vector-space analogue of Kleitman's theorem for all dimensions, establishing exact bounds and extremal configurations, and develops a novel stability theory for subspace diameter problems.

Contribution

It provides the exact bounds and extremal configurations for the vector-space analogue of Kleitman's theorem across all parameters, and introduces a new stability framework for subspace diameter issues.

Findings

Resolved the sharp bounds for all n ≥ d+1 in the vector-space analogue of Kleitman's theorem.

Established the structure of extremal configurations and stability properties in the vector space setting.

Discovered a parity-based split in stability behavior between even and odd diameters.

Abstract

In 1966, Kleitman determined the maximum size of a family of subsets of $[n]$ with bounded symmetric difference. Liao, Liu and Yan recently established a vector-space analogue in the cases $n=d+1$ and $n>2d$, and asked for the sharp bound in the remaining range. We resolve this problem completely by proving the exact vector-space analogue of Kleitman's theorem for every $n\ge d+1$, and we also determine all extremal configurations. We further develop a stability theory for the vector-space diameter problem. Unlike the Boolean cube, the lattice of subspaces has no translation symmetry, and this makes the stability theory substantially different from its classical counterpart. The geometry of subspace balls leads to two natural notions: canonical stability, which forbids containment only in the canonical extremal configurations, and global stability, which forbids containment in arbitrary balls or adjacent double balls of the corresponding radius. We determine sharp canonical stability in even diameter, sharp canonical and global stability in odd diameter, and prove a nontrivial general upper bound for global stability in even diameter. In particular, these two notions exhibit a sharp parity split: in odd diameter they collapse to the same problem, whereas in even diameter they lead to genuinely different extremal behavior.