  A graph $G$ is said to be $k$-subspace choosable over a field $\mathbb{F}$ if
for every assignment of $k$-dimensional subspaces of some finite-dimensional
vector space over $\mathbb{F}$ to the vertices of $G$, it is possible to choose
for each vertex a nonzero vector from its subspace so that adjacent vertices
receive orthogonal vectors over $\mathbb{F} $. The subspace choice number of
$G$ over $\mathbb{F}$ is the smallest integer $k$ for which $G$ is $k$-subspace
choosable over $\mathbb{F}$. This graph parameter, introduced by Haynes, Park,
Schaeffer, Webster, and Mitchell (Electron. J. Comb., 2010), is inspired by
well-studied variants of the chromatic number of graphs, such as the (color)
choice number and the orthogonality dimension.
  We study the subspace choice number of graphs over various fields. We first
prove that the subspace choice number of every graph with average degree $d$ is
at least $\Omega(\sqrt{d/\ln d})$ over any field. We then focus on bipartite
graphs and consider the problem of estimating, for a given integer $k$, the
smallest integer $m$ for which the subspace choice number of the complete
bipartite graph $K_{k,m}$ over a field $\mathbb{F}$ exceeds $k$. We prove upper
and lower bounds on this quantity as well as for several extensions of this
problem. Our results imply a substantial difference between the behavior of the
choice number and that of the subspace choice number. We also consider the
computational aspect of the subspace choice number, and show that for every $k
\geq 3$ it is $\mathsf{NP}$-hard to decide whether the subspace choice number
of a given bipartite graph over $\mathbb{F}$ is at most $k$, provided that
$\mathbb{F}$ is either the real field or any finite field.
