Let $S=\{c\_1,\dots,c\_n\}$ be a set of vectors in $\mathbb{R}^M$. Is the below problem studied in literature?  

$$\max\limits\_{S'\subset S} \vert S' \vert $$
$$s.t. dim(span(S')) < dim(span(S))$$


which is to find the cardinality of the largest subset which does not span the span of S.



Largest subset not spanning the span
This seems to be a matroid theory question. If you let $S$ be the ground set of the matroid and you let $r$ be the rank of the matroid (i.e. $\dim \mathrm{span}(S)$), then your question amounts to finding the largest flat of rank $r-1$.


For reference, I would recommend Oxley's Matroid Theory.


That said, one generally cares more about finding the entire lattice of flats or a maximal chain of flats than a particular one. But to answer your question, flats, in general, are of particular interest in matroid theory. There are a number of tools for finding flats as well.


