Exercise 7, § 7

Let $S$ be a non-empty finite subset of a vector $K$-space $E$, not containing 0. Suppose that there exists an integer $k \geqslant 1$ such that for every subset

T of S, Card(T) $\leq k \cdot \mathrm{rg}(T)$. Show that there exists a partition of S into h free subsets for some integer $h \leq k$. (Prove that S satisfies condition (***) of Exercise 6, arguing by induction on Card(S) and taking amongst all the subspaces $F_{r_1 \ldots r_m}$ one of those with the smallest possible dimension; if G is such a subspace and j an index for which Card(G $\cap$ N_j) is the least possible, prove that