Helly numbers of acyclic families

TL;DR

This paper establishes a new upper bound on the Helly number for families of sets in topological spaces based on their connected components and homology properties, improving understanding in geometric transversal theory.

Contribution

It introduces a general bound on Helly numbers for acyclic families in topological spaces, extending previous results and providing explicit bounds in geometric transversal theory.

Findings

Bound on Helly number: at most r(d_Gamma+1).

The bound is proven to be optimal.

Improves bounds in geometric transversal theory.

Abstract

The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Gamma. Assume that for every sub-family G of F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set of Gamma has trivial Q-homology in dimension j and higher. (In particular d_{R^d} = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known.