  Let $k,d,\lambda\geqslant1$ be integers with $d\geqslant\lambda $. Let
$m(k,d,\lambda)$ be the maximum positive integer $n$ such that every set of $n$
points (not necessarily in general position) in $\mathbb{R}^{d}$ has the
property that the convex hulls of all $k$-sets have a common transversal
$(d-\lambda)$-plane. It turns out that $m(k, d,\lambda)$ is strongly connected
with other interesting problems, for instance, the chromatic number of Kneser
hypergraphs and a discrete version of Rado's centerpoint theorem. In the same
spirit, we introduce a natural discrete version $m^*$ of $m$ by considering the
existence of complete Kneser transversals. We study the relation between them
and give a number of lower and upper bounds of $m^*$ as well as the exact value
in some cases. The main ingredient for the proofs are Radon's partition theorem
as well as oriented matroids tools. By studying the alternating oriented
matroid we obtain the asymptotic behavior of the function $m^*$ for the family
of cyclic polytopes.
