Dömötör Pálvölgyi: Radon implies Tverberg, Hellies and weak eps-nets

We define an abstraction of convex sets, called convexity space, and show that if the natural generaliation of Radon's theorem holds, then so does the Tverberg theorem, Helly's theorem, the colorful Helly theorem, the fractional Helly theorem, and thus there are also weak eps-nets. We also study the so-called Eckhoff's conjecture about the growth rate of r_k, the k
-th partition number aka k-th Radon number aka k-th Tverberg number.

Bibliogaphy:
A. F. Holmsen and Dong-Gyu Lee: Radon numbers and the fractional Helly theorem, arXiv.
B. Bukh: Radon partitions in convexity spaces, arXiv.
A. F. Holmsen: Large cliques in hypergraphs with forbidden substructures, arXiv.