Helly type problems in convexity spaces

Abstract

We report on some recent progress regarding combinatorial properties in convexity spaces with a bounded Radon number. In particular, we discuss the relationship between the Radon number, the colorful and fractional Helly properties, weak $\varepsilon$-nets, and $(p,q)$-theorems. As an application of the theory of convexity spaces we introduce new classes of uniform hypergraphs and show that they are $\chi$-bounded.