Bounding Radon Numbers via Betti Numbers

We prove general topological Radon-type theorems for sets in $\mathbb R^{d}$ or on a surface. Combined with a recent result of Holmsen and Lee, we also obtain fractional Helly theorem, and consequently the existence of weak $\varepsilon $-nets as well as a $(p,q)$-theorem for those sets. More precisely, given a family ${\mathcal{F}}$ of subsets of ${\mathbb{R}}^{d}$, we will measure the homological complexity of ${\mathcal{F}}$ by the supremum of the first $\lceil d/2\rceil $ reduced Betti numbers of $\bigcap{\mathcal{G}}$ over all nonempty ${\mathcal{G}} \subseteq{\mathcal{F}}$. We show that if ${\mathcal{F}}$ has homological complexity at most $b$, the Radon number of ${\mathcal{F}}$ is bounded in terms of $b$ and $d$. In case that ${\mathcal{F}}$ lives on a surface and the number of connected components of $\bigcap \mathcal G$ is at most $b$ for any $\mathcal G\subseteq \mathcal F$, then the Radon number of ${\mathcal{F}}$ is bounded by a function depending only on $b$ and the surface itself. For surfaces, if we moreover assume the sets in ${\mathcal{F}}$ are open, we show that the fractional Helly number of $\mathcal F$ is linear in $b$. The improvement is based on a recent result of the author and Kalai. Specifically, for $b=1$ we get that the fractional Helly number is at most three, which is optimal. This case further leads to solving a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.