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Further Consequences of the Colorful Helly Hypothesis

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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.

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Notes

  1. This can be attributed to the integer coefficients of the variables \(\{f(e)\}_{e\in {{\mathcal {E}}}_{\mathcal {A}}}\) and \(\{g(e)\}_{e\in {{\mathcal {E}}}_{\mathcal {B}}}\) in the linear inequalities that “delimit” the domain; see, e.g., [19, Thm. 10.1.1, Chap. 10].

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Acknowledgements

The authors thank the anonymous SoCG and DCG referees for valuable comments which helped to improve the presentation. An extended abstract of this paper has appeared in Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018). The project leading to this application has received funding from European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement No. 678765. The first and third authors were also supported by Grant 1452/15 from Israel Science Foundation. The third author was also supported by Ralph Selig Career Development Chair in Information Theory and Grant 2014384 from the U.S.–Israeli Binational Science Foundation. The second author was supported by PAPIIT Project IA102118.

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  1. Leonardo Martínez-Sandoval

    Present address: Institut de Mathématiques de Jussieu - Paris Rive Gauche (UMR 7586), 4 Place Jussieu, 75252, Paris Cedex 5, France

Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University of the Negev, 84105, Beer-Sheva, Israel

    Leonardo Martínez-Sandoval & Natan Rubin

  2. Centro de Ciencias Matemáticas, UNAM campus Morelia, 58190, Morelia, Michoacán, Mexico

    Edgardo Roldán-Pensado

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  1. Leonardo Martínez-Sandoval
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  2. Edgardo Roldán-Pensado
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Correspondence to Natan Rubin.

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Martínez-Sandoval, L., Roldán-Pensado, E. & Rubin, N. Further Consequences of the Colorful Helly Hypothesis. Discrete Comput Geom 63, 848–866 (2020). https://doi.org/10.1007/s00454-019-00085-y

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