Helly type problems in convexity spaces

TL;DR

This paper explores combinatorial properties in convexity spaces with bounded Radon numbers, establishing connections between various Helly-type properties and introducing new hypergraph classes that are chi-bounded.

Contribution

It advances the understanding of convexity spaces by linking Radon numbers with Helly properties and introduces new chi-bounded hypergraph classes.

Findings

Radon number influences Helly-type properties

Established relationships between fractional and colorful Helly properties

Introduced new chi-bounded hypergraph classes

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.