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The lonely runner with seven runners. (English) Zbl 1206.11030

Summary: Suppose \(k+1\) runners having nonzero constant pairwise distinct 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 (\(k\leq 5\)). 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.

MSC:

11B75 Other combinatorial number theory
11J71 Distribution modulo one
05C15 Coloring of graphs and hypergraphs
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