The lonely runner with 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 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.
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