Discrete convexity : retractions, morphisms and the partition problem

Introducing certain types of morphisms for general (abstract) convexity spaces, we give several ways for reducing the (Calder-) Eckhoff partition problem to simpler equivalent forms (finite, point-convex, interval spaces; restricted form, i.e. with distinct points). With additional results (to appear in a forthcoming paper) we show how the general problem can be reduced to its restricted version for finite graph-geodetic convexities (in a connected graph a set of vertices is geodetically convex if it contains the vertices of any shortest path joining two of its elements). For instance, for every convexity space with Helly number h and partition numbers pk≥4, we provide (constructively) a finite connected graph whose (geodetic) convexity has Helly number h, Radon number p2 and a k-th partition number not less than pk. Mathematics Subject Classification (MSC) 1991 *52A01 Axiomatic and generalized geometric convexity *05C99 Graph theory 52A30 Variants of convex sets 54E35 Metric spaces, metrizability