Tverberg-Type Theorems for Separoids

Let S be a d-dimensional separoid of (k− 1)(d+1)+1 convex sets in some ‘large-dimensional’ Euclidean space IE . We prove a theorem that can be interpreted as follows: if the separoid S can be mapped with a monomorphism to a d-dimensional separoid of points P in general position, then there exists a k-colouring ς:S → Kk such that, for each pair of colours i, j ∈ Kk, the convex hulls of their preimages do intersect —they are not separated. Here, by a monomorphism we mean an injective function such that the preimage of separated sets are separated. In a sense, this result is ‘dual’ to the Hadwiger-type