Coloring the Real Line with Monochromatic Intervals

TL;DR

The paper constructs specific distance sets D to show that for any number of colors t>1, there exists a graph with chromatic number t that cannot be properly colored with monochromatic intervals, disproving a previous conjecture.

Contribution

It demonstrates the existence of distance sets D leading to graphs with prescribed chromatic numbers that defy monochromatic interval colorings, challenging prior assumptions.

Findings

For any t>1, there exists a distance set D with chromatic number t.

Such graphs cannot be properly colored with monochromatic intervals.

This disproves the conjecture in prior work.

Abstract

Let D be a finite set of positive real numbers. The distance graph G(R,D) is the graph with vertex set R (set of real numbers), and two vertices x, y are adjacent if |x-y| belongs to D. We prove that every positive integer t>1 there is a distance set D such that the chromatic number of G(R,D) is t and no proper coloring of G(R,D) with t colors allows monochromatic intervals. This result disproves a conjecture in [2].