A fractional Helly theorem for set systems with slowly growing homological shatter function

TL;DR

This paper establishes a fractional Helly theorem for set systems in Euclidean space with bounded Betti numbers for intersections, generalizing previous results by introducing graded Radon and Helly numbers.

Contribution

It introduces graded Radon and Helly numbers and proves a fractional Helly theorem for convexity spaces with bounded homological shatter functions, extending known theorems.

Findings

Families satisfy fractional Helly theorem despite unbounded Radon number

Introduction of graded Radon and Helly numbers

Generalization of existing fractional Helly theorems

Abstract

We study parameters of the convexity spaces associated with families of sets in $\mathbb{R}^d$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$. Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems.