Tverberg numbers for cellular bipartite graphs

(The case n = 2 was settled by Birch [4]. For new proofs of the Tverberg theorem see [13] and [17].) These theorem of Radon and Tverberg are formulated completely in ~erms of intersections of convex hulls and this suggests to formulate corresponding statements for more general kinds of convexities. A family cg of subsets of a set X is called a convexi ty on X if cg contains 0 and X and is closed under arbitrary intersections and directed unions; see Soltan [14] and van de Vel [18]. The members of cg are called convex sets. The convex hull co (S) of any subset S of X is defined to be the intersection of all convex sets which contain S. For instance, every connected graph m endowed with the geodesic convexity , consisting of all those subsets S of the vertex set which include each shortest path of G joining two vertices of S. As to the definition of classical convex invariants [such as the Helly, Carath6dory, Radon and Tverberg numbersl of a convexity we adopt the convention of van de Vel [18]. The Radon number of a convexity ~ is the smallest integer r (if it exists) such that any finite set S with I SI > r admits a partition {$1, S 2 with