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

2024/11/27 by Marguerite Bin, Bin, Marguerite
Computer Science · Mathematics · #Computational Geometry (cs.CG) #FOS: Computer and information sciences #Functional Equations Stability Results #Polynomial and algebraic computation #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2411.18605

openalex publication_date 2024/11/27 · openalex created_date 2024/12/05 · openalex updated_date 2026/07/28

Abstract

We study parameters of the convexity spaces associated with families of sets in 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.

Related

• The fractional Helly number for separable convexity spaces
• Radon numbers and the fractional Helly theorem
• Helly type problems in convexity spaces
• Radon numbers grow linearly
• Transversal numbers over subsets of linear spaces