Type: Article

Further Consequences of the Colorful Helly Hypothesis

Martinez-Sandoval, L; Roldan-Pensado, E; Rubin, N;

Abstract:

Let F be a family of convex sets in R-d, which are colored with d + 1 colors. We say that F satisfies the Colorful Helly Property if every rainbow selection of d + 1 sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovasz states that for any such colorful family F there is a color class F-i subset of F, for 1 <= i <= d + 1, whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension d >= 2 there exist numbers f (d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f (d) points, or all the sets in F can be crossed by g(d) lines.