  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).
