# Radon numbers and the fractional Helly theorem

## TL;DR

This paper establishes a fractional Helly theorem for convexity spaces with bounded Radon numbers, leading to a weak epsilon-net theorem, thereby advancing understanding of combinatorial complexity in convexity spaces.

## Contribution

It proves a fractional Helly theorem for convexity spaces with bounded Radon numbers, answering longstanding questions and extending recent results.

## Key findings

- Established a fractional Helly theorem for convexity spaces with bounded Radon number.
- Derived a weak epsilon-net theorem for these convexity spaces.
- Extended previous results by Moran and Yehudayoff.

## Abstract

A basic measure of the combinatorial complexity of a convexity space is its Radon number. In this paper we show a fractional Helly theorem for convexity spaces with a bounded Radon number, answering a question of Kalai. As a consequence we also get a weak epsilon-net theorem for convexity spaces with a bounded Radon number. This answers a question of Bukh and extends a recent result of Moran and Yehudayoff.