The colorful Helly theorem and general hypergraphs

AbstractThe definition of the Helly property for hypergraphs was motivated by the Helly theorem for convex sets. Similarly, we define the colorful Helly property for a family of hypergraphs, motivated by the colorful Helly theorem for collections of convex sets, by Lovász. We describe some general facts about the colorful Helly property and prove complexity results. In particular, we show that it is Co-NP-complete to decide if a family of p hypergraphs is colorful Helly, even if p=2. However, for any fixed p, we describe a polynomial time algorithm to decide if such family is colorful Helly, provided at least p−1 of the hypergraphs are p-Helly

Quantitative Fractional Helly and (p,q)(p,q)-Theorems

We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the $(p,q)$-Theorem of Alon and Kleitman.Comment: 11 page