Colorful Helly via induced matchings

TL;DR

This paper links the maximum size of induced matchings in bipartite complements of incidence graphs to bounds on the colorful Helly number, providing new insights into combinatorial set systems.

Contribution

It establishes a novel upper bound on the colorful Helly number using induced matchings in bipartite complements of incidence graphs.

Findings

Maximum induced matching size bounds the colorful Helly number

Provides a new combinatorial bound for set systems

Discusses refinements and applications of the main theorem

Abstract

We establish a theorem regarding the maximum size of an {\it{induced}} matching in the bipartite complement of the incidence graph of a set system $(X,\mathcal{F})$. We show that this quantity plus one provides an upper bound on the colorful Helly number of this set system, i.e. the minimum positive integer $N$ for which the following statement holds: if finite subfamilies $\mathcal{F}_1,\ldots, \mathcal{F}_{N} \subset \mathcal{F}$ are such that $\cap_{F \in \mathcal{F}_{i}} F = 0$ for every $i=1,\ldots,N$, then there exists $F_i \in \mathcal{F}_i$ such that $F_1 \cap \ldots \cap F_{N} = \emptyset$. We will also discuss some natural refinements of this result and applications.