Regular chromatic number and the lonely runner problem

Suppose k + 1 runners having nonzero constant speeds run laps on a unit-length circular track. The lonely runner conjecture states that there is a time at which one runner is at distance at least 1 / ( k + 1 ) from all the others. 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.