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Summary: Splitting necklaces and measurable colorings of the real line
Noga Alon, Jaroslaw Grytczuk, Michal LasoŽn, and Mateusz Michalek
Abstract. A (continuous) necklace is simply an interval of the real line col-
ored measurably with some number of colors. A well-known application of
Borsuk-Ulam theorem asserts that every k-colored necklace can be fairly split-
ted 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
can be fairly splitted 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 in 1991 by Keršanen. Curiously, in
the discrete case the desired coloring also uses four colors.
1. Introduction
In 1906 Thue [23] proved that there is a 3-coloring of the integers such that
no two adjacent intervals are colored exactly the same. This result has lots of
unexpected applications in distinct areas of mathematics and theoretical computer
science (cf. [1], [6], [8], [19]). Many variations and generalizations of this property
were considered so far, specifically in other combinatorial settings like Euclidean
spaces [6], [14], [15], or graph colorings [3], [4], [16].
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