A Radon theorem for Helly graphs (Q1123404)

A graph G is called a Helly graph if every family of metric balls intersecting in pairs has a nonempty intersection; a metric ball with center u and radius k is the set of vertices v such that d(u,v)\(\leq k\) with d a distance function of G. The following theorem is proved. Let G be a Helly graph. Let \(4\leq | X| <\infty\) such that for each partition \(X=Y{\dot \cup}Z\) with \(1\leq | Y| \leq 2\) the convex hulls of Y and Z do not intersect. Then there is a bijection \(x\to x'\) from X to a clique K of G such that for any two distinct vertices \(y,z\in X\) their images \(y'\) and \(z'\), respectively, lie on a shortest path between y and z. Therefore, the Radon number of the geodesic convexity of G (A is called geodesically convex if all shortest paths of G joining two vertices of A belong to A) equals the clique number \(\omega\) of G, provided that \(2\leq | \omega | \leq \infty\). The question whether Helly graphs support Eckhoff's conjecture for generalized Radon numbers is an open problem [see the reviewer and \textit{J. Ch. Boland}, J. Geom. 20, 116-121 (1983; Zbl 0516.52005)].