Avoiding Brooms, Forks, and Butterflies in the Linear Lattices

TL;DR

This paper characterizes the largest families of subspaces in the lattice of subspaces over a finite field that avoid certain small poset configurations, extending classical results with new structural insights.

Contribution

It strengthens known maximal anti-chain results by classifying maximum families avoiding specific subposets like $ abla$, $ riangle$, brooms, forks, and butterflies, with a near-complete uniqueness result.

Findings

Maximum-sized families avoiding $ abla$ and $ riangle$ are the middle level subspaces, except in one case.

The union of two adjacent levels forms the largest family avoiding a butterfly.

Results extend to families avoiding brooms and forks, with uniqueness in most cases.

Abstract

Let $n$ be a positive integer, $q$ a power of a prime, and $\mathcal{L}_n(q)$ the poset of subspaces of an $n$-dimensional vector space over a field with $q$ elements. This poset is a normalized matching poset and the set of subspaces of dimension $\left\lfloor n/2 \right\rfloor$ or those of dimension $\left\lceil n/2 \right\rceil$ are the only maximum-sized anti-chains in this poset. Strengthening this well-known and celebrated result, we show that, except in the case of $\mathcal{L}_3(2)$, these same collections of subspaces are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid both a $\wedge$ and a $\vee$ as a subposet. We generalize some of the results to brooms and forks, and we also show that the union of the set of subspaces of dimension $k$ and $k+1$, for $k = \left\lfloor n/2 \right\rfloor$ or $k = \left\lceil n/2 \right\rceil-1$, are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid a butterfly (definitions below).