Kneser Transversals

Let X be a set of n points in Rd. We are interested in studying whether there is a transversal (d−λ)-plane to the convex hulls of all k-subsets of X. In particular, we would like to know how small n must be to ensure that there will always be such a transversal. This problem has interesting and deep connections with classic problems as Rado's central affine plane problem and the calculation of the chromatic number of Kneser hypergraphs.