Helly-type theorems for abstract convexity spaces

Abstract

In the standard d-dimensional Euclidean convexity space, it is known that the colorful Helly number and the fractional Helly number are both d+1. We prove that in an abstract convexity space with bounded Radon number, the colorful Helly number and the weak fractional Helly number are bounded. To prove these results, we define convexity invariants for abstract simplicial complexes and show several correspondences between convexity invariants of convexity spaces and convexity invariants of nerve complexes. Furthermore, we show a simplification of Bukh’s counterexample for Eckhoff's conjecture as another application of these correspondences.