## Splitting necklaces and measurable colorings of the real line

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- by Noga Alon, Jarosław Grytczuk, Michał Lasoń and Mateusz Michałek
- Proc. Amer. Math. Soc.
**137**(2009), 1593-1599 - DOI: https://doi.org/10.1090/S0002-9939-08-09699-8
- Published electronically: November 25, 2008
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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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## Bibliographic 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
- MR Author ID: 25060
- 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 Lasoń**- 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
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**137**(2009), 1593-1599 - MSC (2000): Primary 05C38, 15A15; Secondary 05A15, 15A18
- DOI: https://doi.org/10.1090/S0002-9939-08-09699-8
- MathSciNet review: 2470817