Let $w_1, w_2,....w_M$ be a set of vectors. There exists no pair of parallel vectors and $D$ vectors are linearly independent. Let $C_k$ be the number of $k$-tuples of linearly independent vectors. I mean that each $k$-tuple lies on a different vectorial subspace. Then I want to find an upper limit for $C_k$, given $C_{k+1}$.

In particular, I would be interested to the case that $D$ is smaller than $M/2$.

k-tuple lies on a different vectorial subspace" contradicts the definition of $C_k$. E.g. with vectors (0,1), (1,0) and (1,1), then $C_2=3$ (or I guess $C_2=6$ since we're considering ordered pairs), but all pairs of vectors span the same space $R^2$. – Thierry Zell Dec 13 '10 at 0:35