Transversals to the convex hulls of all k-sets of discrete subsets of Rn

Let k,d,λ⩾1 be integers with d⩾λ. What is the maximum positive integer n such that every set of n points in Rd has the property that the convex hulls of all k-sets have a transversal (d−λ)-plane? What is the minimum positive integer n such that every set of n points in general position in Rd has the property that the convex hulls of all k-sets do not have a transversal (d−λ)-plane? In this paper, we investigate these two questions. We define a special Kneser hypergraph and, by using some topological results and the well-known λ-Helly property, we relate our second question to the chromatic number of such hypergraphs. Moreover, we establish a connection (when λ=1) with Kneser's conjecture, first proved by Lovász. Finally, we prove a discrete flat center theorem.