## Splitting necklaces and measurable colorings of the real line

### Authors:

- Noga Alon,
- Jarosław Grytczuk,
- Michał Lasoń,
- Mateusz Michałek

### 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.

- Record ID
- WUT29fd44ff174446879daf542c3587d04b
- Author
- Journal series
- Proceedings of the American Mathematical Society, ISSN 0002-9939
- Issue year
- 2009
- Vol
- 137
- No
- 05
- Pages
- 1593-1599
- Publication size in sheets
- 0.50
- ASJC Classification
- ;
- DOI
- DOI:10.1090/S0002-9939-08-09699-8 Opening in a new tab
- URL
- http://www.ams.org/journals/proc/2009-137-05/S0002-9939-08-09699-8/S0002-9939-08-09699-8.pdf Opening in a new tab
- Language
- (en) English
- Score (nominal)
- 0
- Score source
- journalList
- Publication indicators
- = 4; = 4; : 2014 = 1.093; : 2009 = 0.640 (2) - 2009=0.668 (5)
- Citation count
- 5

- Uniform Resource Identifier
- https://repo.pw.edu.pl/info/article/WUT29fd44ff174446879daf542c3587d04b/

- URN
`urn:pw-repo:WUT29fd44ff174446879daf542c3587d04b`

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or PerishOpening in a new tab system.