Splitting necklaces and measurable colorings of the real line

TL;DR

This paper constructs a measurable coloring of the real line with four colors that prevents fair splitting of any interval with a limited number of cuts, contrasting with classical results for necklaces.

Contribution

It introduces a measurable (k+3)-coloring of the real line that blocks fair splitting with at most k cuts, extending discrete results to the continuous setting.

Findings

Existence of a 4-coloring of the real line with no fair split intervals.

Contrast with classical necklace splitting results using the Borsuk-Ulam theorem.

Analogous discrete problem solved with four colors.

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 Erd\H{o}s in 1961 and solved in the affirmative in 1991 by Ker\"anen. Curiously, in the discrete case the desired coloring also uses four colors.