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Electronic Journal of Combinatorics
Article . 2008 . Peer-reviewed
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zbMATH Open
Article . 2008
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https://dx.doi.org/10.48550/ar...
Article . 2007
License: arXiv Non-Exclusive Distribution
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Article . 2020
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The Lonely Runner with Seven Runners

The lonely runner with seven runners
Authors: Javier Barajas; Oriol Serra;

The Lonely Runner with Seven Runners

Abstract

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 ($k\le 5$). 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.

Keywords

Coloring of graphs and hypergraphs, Mathematics - Number Theory, Other combinatorial number theory, FOS: Mathematics, Mathematics - Combinatorics, 11B75, 11J71, Combinatorics (math.CO), Number Theory (math.NT), Distribution modulo one, view obstruction problems

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    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    18
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Top 10%
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Top 10%
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
18
Top 10%
Top 10%
Average
Green
gold