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 Lovász states that for any such colorful family F there is a color class F_i subset 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.
@InProceedings{martinezsandoval_et_al:LIPIcs.SoCG.2018.59, author = {Mart{\'\i}nez-Sandoval, Leonardo and Rold\'{a}n-Pensado, Edgardo and Rubin, Natan}, title = {{Further Consequences of the Colorful Helly Hypothesis}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {59:1--59:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.59}, URN = {urn:nbn:de:0030-drops-87726}, doi = {10.4230/LIPIcs.SoCG.2018.59}, annote = {Keywords: geometric transversals, convex sets, colorful Helly-type theorems, line transversals, weak epsilon-nets, transversal numbers} }
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