Further consequences of the colorful Helly hypothesis. (English) Zbl 1472.52011

Summary: Let \(\mathcal{F}\) be a family of convex sets in \(\mathbb{R}^d,\) which are colored with \(d + 1\) colors. We say that \(\mathcal{F}\) 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 Lovász states that for any such colorful family \(\mathcal{F}\) there is a color class \(\mathcal{F}_i \subset \mathcal{F},\) for \(1 \le i \le d + 1,\) whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension \(d \ge 2\) 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 \(\mathcal{F}\) can be crossed by \(g(d)\) lines.


52A35 Helly-type theorems and geometric transversal theory
52C45 Combinatorial complexity of geometric structures
05D15 Transversal (matching) theory
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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