Let $S\subseteq M_{n,m}(\mathbb{C})$ be a $d$-dimensional subspace of the space of $n\times m$ complex matrices (with $n\leq m$, say). I am interested in figuring out conditions on $S$ which guarantee the existence of a full rank matrix in $S$. Since the maximum dimension of a subspace in which every matrix has rank $\leq (n-1)$ is known to be equal to $m(n-1)$ (see the answer to this question and other linked questions), it is clear that if $d>m(n-1)$, then $S$ must contain a full rank matrix. Are other such simple conditions known which ensure the existence of a full rank matrix in $S$?
According to this paper, the $n=m$ case of the stated problem is quite well-known and is known as the Edmond's problem, which has led to exciting developments in the field of matroid theory. Since I'm not familiar with this field, I would love to gather some references for interesting results in this field which are directly related to the mentioned problem. Thanks!
