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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Splitting necklaces and measurable colorings of the real line


Authors: Noga Alon, Jarosław Grytczuk, Michał Lason and Mateusz Michałek
Journal: Proc. Amer. Math. Soc. 137 (2009), 1593-1599
MSC (2000): Primary 05C38, 15A15; Secondary 05A15, 15A18
Published electronically: November 25, 2008
MathSciNet review: 2470817
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Abstract: A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every $ k$-colored necklace can be fairly split by at most $ k$ cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every $ k\geq 1$ there is a measurable $ (k+3)$-coloring of the real line such that no interval can be fairly split using at most $ k$ cuts. In particular, there is a measurable $ 4$-coloring of the real line in which no two adjacent intervals have the same measure of every color. An analogous problem for the integers was posed by Erdős in 1961 and solved in the affirmative by Keränen in 1991. Curiously, in the discrete case the desired coloring also uses four colors.


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Additional Information

Noga Alon
Affiliation: Schools of Mathematics and Computer Science, Raymond and Beverly Sacler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel – and – Institute for Advanced Study, Princeton, New Jersey 08540
Email: nogaa@tau.ac.il

Jarosław Grytczuk
Affiliation: Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-387 Kraków, Poland
Email: grytczuk@tcs.uj.edu.pl

Michał Lason
Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University, 30-387 Kraków, Poland
Email: mlason@op.pl

Mateusz Michałek
Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University, 30-387 Kraków, Poland
Email: wajcha2@poczta.onet.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09699-8
PII: S 0002-9939(08)09699-8
Keywords: Measurable coloring, splitting necklaces
Received by editor(s): March 25, 2008
Received by editor(s) in revised form: July 4, 2008
Published electronically: November 25, 2008
Communicated by: Jim Haglund
Article copyright: © Copyright 2008 American Mathematical Society