If $|L \cap V|$ is a multiple of $r$, we can construct such a set $R_L$ easily by taking a disjoint cover of $r$-tuples. If $|L \cap V|$ is not a multiple of $r$ (but is still of size at least $r$) we can take a maximal set of disjoint $r$-tuples inside it and then add to it one more $r$-tuple that will cover the remaining elements and will otherwise intersect only one other $r$-tuple. This will guarantee that the third condition holds. We define $R \subset {V \choose r}$ to be the union of all sets $R_L$ over all $r$-rich lines $L$. We can now prove:

\[cla-rtuples\] The set $R \subset {V \choose r}$ defined above has the following three properties.

1.  Each point $v \in V$ is contained in at least $k$ $r$-tuples from $R$.

2.  Every pair of distinct points $u,v \in V$ is contained together in at most two $r$-tuples from $R$.

3.  Let $(v_{i_1},\ldots,v_{i_r}) \in R$. Then there exists $r$ non zero coefficients $\alpha_1,\ldots,\alpha_{r} \in \C$ so that $ \alpha_1 \cdot u_{i_1} + \ldots + \alpha_r \cdot u_{i_r} = 0$.

If, in addition, we know that each point belongs to at most $8k$ rich lines 