In this paper some bounds on the Tveberg-type convexity partition numbers of abstract spaces will be presented. The main objective is to show that a conjecture of J. Eckhoff relating the Tverberg numbers to the Radon number is valid for a certain class of spaces which include ordered sets, trees, pairwise products of trees and subspaces of these. (Application of the Main Theorem to a certain class of semilattices is given in an appendix.) For ordered sets the results here improve those of P. W. Bean and are best possible for general ordered sets. 1* Introduction* To establish terminology, recall that an alignment [7, 8] ("algebraic closure system" [2]) on a set X is a family Sf of subsets of X— to be regarded as "convex" subsets—such that (Al) 0,16^, (A2) arbitrary intersections of sets in £f are again in £f, (A3) unions of upward directed families of sets in £f are again in £?. The smallest convex set containing a set S is denoted £f(S) and is called the hull of S. A Radon m-partition of S is a partition of S into m subsets
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