Bounding Radon Numbers via Betti Numbers

Abstract We prove general topological Radon-type theorems for sets in 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 -nets as well as a (p, q) -theorem for those sets. More precisely, given a family F of subsets of R^d, we will measure the homological complexity of F by the supremum of the first d/2 reduced Betti numbers of G over all nonempty G F. We show that if F has homological complexity at most b, the Radon number of F is bounded in terms of b and d. In case that F lives on a surface and the number of connected components of G is at most b for any nonempty G F, then the Radon number of F is bounded by a function depending only on b and the surface itself. For surfaces, if we moreover assume the sets in F are open, we show that the fractional Helly number of 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.