Is it possible to determine (or give bounds for) the following extremal problem: 


Let $k,m,r$ be positive integers such that $k,m \geq r$. What is the least number $n$ such that for any $r \times n$ matrix in which any subset of $k$ columns have full rank $r$, must contain a size $m$ subset of the columns in which any $r$-subset has full rank.



An extremal problem on matrices
We may get a better bound by the greedy algorithm. Choose columns one by one so that any $r$ chosen columns are linearly independent. Assume that $N<m$ columns are chosen and we cannot proceed. Then any other column lies in one of hyperplanes defined by some $r-1$ chosen vectors. Each such hyperplane contains at most $k-r$ new (not chosen yet) vectors, thus we get $n\leqslant N+(k-r)\binom{N}{r-1}$. In other words, if $n\geqslant m+(k-r)\binom{m-1}{r-1}$, we choose at least $m$ vectors. This is tight for $r=1$ and for $r=2$, but I expect that better bounds should hold for greater values of $r$. 


