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 $\E^N$. 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 $\varsigma\colon \ S\to K_k$ such that, for each pair of colours $i,j\in K_k$, 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 theorems proved by Goodman and Pollack (1988) and Arocha et al. (2002). We also introduce $\T(k,d)$, the minimum number $n$ such that all $d$-dimensional separoids of order at least $n$ can be $k$-coloured as before. By means of examples and explicit colourings, we show that for all $k>2$ and $d>0$, \[(k-1)(d+1)+1<\T(k,d)<{k\choose2}(d+1)+1.\] Furthermore, by means of a probabilistic argument, we show that for each $d$ there exists a constant $C=C(d)$ such that for all $k$, $\T(k,d)\leq Ck\log k$.