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

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 R d 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 R d 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 Kneserʹs conjecture, first proved by Lovász. Finally, we prove a discrete flat center theorem.