Abstract
Let k,d,\lambda\ge 1 be integers with d\ge 1. 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-\lambda)-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-\lmabda)-plane? In this paper, we investigate these two questions. We de ne a special Kneser hypergraph and, by using some topological results and the well-known -Helly property, we relate our second question to the chromatic number of such hypergraphs. Moreover, we establish a connection (when \lmabda = 1) with Kneser's conjecture, rst proved by Lovasz. Finally, we prove a discrete at center theorem.