TY - JOUR

T1 - Further Consequences of the Colorful Helly Hypothesis

AU - Martínez-Sandoval, Leonardo

AU - Roldán-Pensado, Edgardo

AU - Rubin, Natan

N1 - Funding Information:
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.
Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - Let F be a family of convex sets in Rd, 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 Lovász states that for any such colorful family F there is a color class Fi⊂ 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.

AB - Let F be a family of convex sets in Rd, 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 Lovász states that for any such colorful family F there is a color class Fi⊂ 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.

KW - Colorful Helly-type theorems

KW - Convex sets

KW - Geometric transversals

KW - Line transversals

KW - Transversal numbers

KW - Weak epsilon-nets

UR - http://www.scopus.com/inward/record.url?scp=85065290706&partnerID=8YFLogxK

U2 - 10.1007/s00454-019-00085-y

DO - 10.1007/s00454-019-00085-y

M3 - Article

AN - SCOPUS:85065290706

SN - 0179-5376

VL - 63

SP - 848

EP - 866

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -