I asked this question on Stack Exchange two weeks ago, and didn't get any answers, so I'm shamelessly reposting it here.

Let $S=\{v_1,\dots, v_n\} \subset V$ be a set of nonzero vectors in a vector space (we can take the field to be algebraically closed). It is easy to prove that if there exists a sequence $S_1,\dots, S_r \subseteq S$ such that

$\dim\text{span}S_j=1$ for all $j=1,\dots, r$.

$S_j \cap S_{j+1} \neq \emptyset$ for all $j=1,\dots, r-1$.

$\bigcup_{j=1}^r S_j=S$.

then $\dim\text{span}S=1$.

Essentially, this gives a way to upper bound $\dim\text{span}S$ in terms of $\dim\text{span}$ of subsets obeying certain properties. My open-ended question is whether there are "interesting" generalizations of this to the case in which we want to upper bound $\dim\text{span}S\leq d$ for some $d\geq 1$, in terms of subsets obeying certain properties.

Of course, there are trivial results in this vein: If every subset of size $\leq n-1$ has dimension $\leq d$ for some $d<n-1$, then $\dim\text{span}S\leq d$. I am looking for something more useful, that hopefully looks something like the above result for $d=1$.