View-obstruction: a shorter proof for 6 lonely runners
The lonely runner conjecture (see, for example, \textit{W. Bienia}, \textit{L. Goddyn}, \textit{P. Gvozdjak}, \textit{A. Sebö} and \textit{M. Tarsi} [J. Comb. Theory, Ser. B 72, No. 1, 1--9 (1998; Zbl 0910.05064)]) can be stated as follows: Let \(v_1,\dots, v_s\) be positive integers. There is a real number \(t\) such that \(\{tv_i\}\in [1/(s+ 1),s/(s+ 1)]\) for all \(i\in \{1,\dots, s\}\). It is well-known that it is true for \(s= 2,3,4,5\). In the present paper, the author gives a new proof of this conjecture for the case \(s= 5\), basing on a careful study of the different congruence classes modulo 6 of the numbers \(v_1,\dots, v_5\). Moreover, the author also gives very simple and elementary proofs for the cases \(s= 2,3,4\).
scientific article; zbMATH DE number 2116258
scientific article