Avoiding Brooms, Forks, and Butterflies in the Linear Lattices

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 $\lfloor n/2\rfloor$ or those of dimension $\lceil n/2\rceil$ 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 $k=\lfloor n/2\rfloor$ or $k=\lceil n/2\rceil -1$, are the only maximum-sized families in ${\mathcal{L}}_n(q)$ that avoid a butterfly (definitions below).

20 pages