TY - GEN
T1 - Further consequences of the colorful helly hypothesis
AU - Martínez-Sandoval, Leonardo
AU - Roldán-Pensado, Edgardo
AU - Rubin, Natan
N1 - Funding Information:
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. Also supported by grant 1452/15 from Israel Science Foundation Also supported by PAPIIT project IA102118 Also supported by grant 1452/15 from Israel Science Foundation, by Ralph Selig Career Development Chair in Information Theory and grant 2014384 from the U.S.-Israeli Binational Science Foundation
Funding Information:
Also supported by grant 1452/15 from Israel Science Foundation 2 Also supported by PAPIIT project IA102118 3 Also supported by grant 1452/15 from Israel Science Foundation, by Ralph Selig Career Development Chair in Information Theory and grant 2014384 from the U.S.-Israeli Binational Science Foundation
Funding Information:
Funding 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.
Publisher Copyright:
© Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, and Natan Rubin; licensed under Creative Commons License CC-BY 34th Symposium on Computational Geometry (SoCG 2018).
PY - 2018/6/1
Y1 - 2018/6/1
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 F i ⊂ 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 F i ⊂ 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 - https://www.scopus.com/pages/publications/85048945787
U2 - 10.4230/LIPIcs.SoCG.2018.59
DO - 10.4230/LIPIcs.SoCG.2018.59
M3 - Conference contribution
AN - SCOPUS:85048945787
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 591
EP - 5914
BT - 34th International Symposium on Computational Geometry, SoCG 2018
A2 - Toth, Csaba D.
A2 - Speckmann, Bettina
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 34th International Symposium on Computational Geometry, SoCG 2018
Y2 - 11 June 2018 through 14 June 2018
ER -