Name
Abstract | ||
|---|---|---|
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. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1007/s00454-019-00085-y | Discrete & Computational Geometry |
Keywords | DocType | Volume |
Geometric transversals, Convex sets, Colorful Helly-type theorems, Line transversals, Weak epsilon-nets, Transversal numbers, 52C17, 52C35, 52C45, 05D15 | Conference | 63 |
Issue | ISSN | Citations |
4 | 0179-5376 | 0 |
PageRank | References | Authors |
0.34 | 3 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Leonardo Martínez-Sandoval | 1 | 4 | 2.22 |
| Edgardo Roldán-Pensado | 2 | 4 | 5.06 |
| Natan Rubin | 3 | 92 | 11.03 |