The Lonely Runner with Seven Runners

Barajas, Javier; Serra, Oriol

doi:10.37236/772

Cited by 28 publications

(28 citation statements)

References 15 publications

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“…The case n = 6 was proved with a long and complicated proof by Bohmann, Holzman and Kleitman [6] in 2001, which was later simplified by Renault [21] in 2004. Finally, the conjecture for seven runners was established by Barajas and Serra [1] in 2008 and it remains open for all integers n ≥ 8. Several other problems related to the Lonely Runner Conjecture have also been profusely studied such as the gap of loneliness [20,12,13,24] or the validity of the conjecture under various hypotheses on the velocities [19,22,2,17,24].…”

Section: Introductionmentioning

confidence: 93%

On the time for a runner to get lonely

Rifford¹

2021

Preprint

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The Lonely Runner Conjecture asserts that if n runners with distinct constant speeds run on the unit circle R/Z starting from 0 at time 0, then each runner will at some time t > 0 be lonely in the sense that she/he will be separated by a distance at least 1/n from all the others at time t. In investigating the size of t, we show that an upper bound for t in terms of a certain number of rounds (which, in the case where the lonely runner is static, corresponds to the number of rounds of the slowest non-static runner) is equivalent to a covering problem in dimension n − 2. We formulate a conjecture regarding this covering problem and prove it to be true for n = 3, 4, 5, 6. We also show that the so-called gap of loneliness in one round, where we have m + 1 runners including one static runner, is bounded from below by 1/(2m − 1) for all integer m ≥ 2.

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Section: Introductionmentioning

confidence: 93%

On the time for a runner to get lonely

Rifford¹

2021

Preprint

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“…The conjecture has received substantial attention in recent decades. The main approach has been to establish the Lonely Runner Conjecture for small values of n; it is now known to hold for n 6 (see [4] for n = 2 and n = 3; [16,5] for n = 4; [6,23] for n = 5; [3] for n = 6). Another appealing avenue of inquiry has been improving the trivial lower bound ML(v 1 , .…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem

Kravitz¹

2021

Combinatorial Theory

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We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number x, let x denote the distance from x to the nearest integer. For each set of positive integer speeds v 1 , . . . , v n , we define the associated maximum loneliness to beThe Lonely Runner Conjecture asserts that ML(v 1 , . . . , v n ) 1/(n + 1) for all choices of v 1 , . . . , v n . We make the stronger conjecture that for each choice of v 1 , . . . , v n , we have either ML(v 1 , . . . , v n ) = s/(ns + 1) for some s ∈ N or ML(v 1 , . . . , v n ) 1/n. This view reflects a surprising underlying rigidity of the Lonely Runner Problem. Our main results are: confirming our stronger conjecture for n 3; and confirming it for n = 4 and n = 6 in the case where one speed is much faster than the rest.

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“…Motzkin's problem also has connections with some other problems, such as problems related to the fractional chromatic number of distance graphs and the Lonely Runner Conjecture. One can refer to [9,[20][21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

On Optimal M-Sets Related to Motzkin’s Problem

Yang

Pan

2020

Journal of Mathematics

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Let M be a set of positive integers. A set S of nonnegative integers is called an M ‐ set if a and b ∈ S , then a − b ∉ M . If S ⊆ 0,1 , … , n is an M − set with the maximal cardinality, then S is called a maximal M − set of 0,1 , … , n . If S ∩ 0,1 , … , n is a maximal M − set of 0,1 , … , n for all integers n ≥ 0 , then we call S an optimal M − set. In this paper, we study the existence of an optimal M − set.

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The Lonely Runner with Seven Runners

Cited by 28 publications

References 15 publications

On the time for a runner to get lonely

On the time for a runner to get lonely

Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem

On Optimal M-Sets Related to Motzkin’s Problem

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