The lonely runner with seven runners

TL;DR

This paper proves the first open case of the lonely runner conjecture for seven runners using a new tool related to the chromatic number of distance graphs, advancing understanding of this longstanding problem.

Contribution

The paper introduces a novel method based on distance graph chromatic numbers to resolve the lonely runner conjecture for seven runners, the first unresolved case.

Findings

Confirmed the conjecture for seven runners

Developed a new tool for analyzing distance graphs

Extended the known cases of the conjecture to seven runners

Abstract

Suppose $k+1$ runners having nonzero constant speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least $1/(k+1)$ along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners ($k\le 5$). A formulation of the problem is related to the regular chromatic number of distance graphs. We use a new tool developed in this context to solve the first open case of the conjecture with seven runners.