SPLITTING NECKLACES AND MEASURABLE COLORINGS OF THE REAL LINE

Noga Alon*, Jaroslaw Grytczuk, Michal Lason, Mateusz Michalek

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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 >= 1 there is a measurable (k+3)-coloring of the real line such that no interval call 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 Erdos in 1961 and solved in the affirmative by Keranen in 1991. Curiously, in the discrete case the desired coloring also uses four colors.

Original languageEnglish
Pages (from-to)1593-1599
Number of pages7
JournalProceedings of the American Mathematical Society
Volume137
Issue number5
Publication statusPublished - 2009
MoE publication typeA1 Journal article-refereed

Keywords

  • Measurable coloring
  • splitting necklaces
  • NON-REPETITIVE SEQUENCES
  • NONREPETITIVE COLORINGS
  • BISECTION
  • SYMBOLS
  • GRAPHS
  • SETS

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