Radon theorem plays a basic role in many results of combinatorial convexity. It says that any set of $d+2$ points in $\mathbb{R}^d$ 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 $\mathscr{F}$ of subsets of $\mathbb{R}^d$, we will measure the complexity of $\mathscr{F}$ by the supremum of the first $d/2$ Betti numbers over all elements of $\mathscr F$. We show that bounded complexity of $\mathscr{F}$ guarantees versions of all the results mentioned above.
The talk is partially based on joint work with Xavier Goaoc and Andreas Holmsen.