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

Barajas, J.; Serra, O.

Electron. J. Comb. 15, No. 1, Research Paper R48, 18 p. (2008).

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

Keywords:

view obstruction problems