Radon theorem plays a basic role in many results of combinatorial convexity. It says that any set of d+2 points in R^d can be split into two parts so that their convex hulls intersect. It implies Helly theorem and as shown recently also its more robust version, so-called fractional Helly theorem. By standard techniques, this consequently yields an existence of weak epsilon nets and a (p,q)-theorem.

We will show that we can obtain these results even without assuming convexity, replacing it with very weak topological conditions. More precisely, given an intersection-closed family F of subsets of R^d, we will measure the complexity of F by the supremum of the first d/2 Betti numbers over all elements of F. We show that bounded complexity of F guarantees versions of all the results mentioned above.

Partially based on joint work with Xavier Goaoc and Andreas Holmsen.