Colorful Helly via induced matchings

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 $$\mathcal{F}<em>1,\ldots, \mathcal{F}</em>{N} \subset \mathcal{F}$$ are such that $\cap_{F \in \mathcal{F}<em>{i}} F = 0$ for every $i=1,\ldots,N$, then there exists $F_i \in \mathcal{F}_i$ such that $F_1 \cap \ldots \cap F</em>{N} = \emptyset$. We will also discuss some natural refinements of this result and applications.