The fractional Helly number for separable convexity spaces

TL;DR

The paper extends the fractional Helly theorem from convex lattice sets to separable convexity spaces with a bounded Radon number by leveraging the dual VC-dimension of halfspaces. The authors prove that, under separability and a dual VC-dimension bound $d$ for the halfspaces, the fractional Helly number is at most $d+1$, with corollaries giving a universal bound of $2^r$ tied to the Radon number $r$. The approach combines a weak colorful Helly result with Erdős–Simonovits supersaturation to pass from many intersecting $(d+1)$-tuples to many intersecting $m$-tuples and then applies a standard fractional Helly theorem. These results unify and extend known cases (e.g., Euclidean convex sets, convex lattice sets) and clarify the role of separation axioms and VC-dimension in fractional Helly theory, while also providing a counterexample to a Bárány–Kalai conjecture on polynomial-inequality solutions.

Abstract

A convex lattice set in $\mathbb{Z}^d$ is the intersection of a convex set in $\mathbb{R}^d$ and the integer lattice $\mathbb{Z}^d$. A well-known theorem of Doignon states that the Helly number of $d$-dimensional convex lattice sets equals $2^d$, while a remarkable theorem of Bárány and Matoušek states that the fractional Helly number is only $d+1$. In this paper we generalize their result to abstract convexity spaces that are equipped with a suitable separation property. We also disprove a conjecture of Bárány and Kalai about an existence of fractional Helly property for a family of solutions of bounded-degree polynomial inequalities.

Theorems & Definitions (15)

• Theorem 1: Bárány--Matoušek
• Theorem 2: Levi
• Theorem 3: Holmsen--Lee
• Theorem 4: Matoušek
• Theorem 5
• Corollary 6
• Proposition 7
• proof
• Proposition 8
• Proposition 9: Erdős--Simonovits