Inquiries about lonely runner conjecture

[Einstein's Cat]

Here is the Wikipedia article on the lonely runner conjecture

https://en.m.wikipedia.org/wiki/Lonely_runner_conjecture#

I have some inquires about it.

Firstly I am right in thinking that "pair wise distinct" means that the speed of all runners are different?

Also does every runner have to be at least 1/k (k is the number of runners) from all the other runners simultaneously? Or can runners achieve "loneliness" at a certain time even whilst other runners have not?

Furthermore, once a runner achieves "loneliness" is it maintained? In other words say a runner is lonely, does it stay lonely even if the distance between it and the other runners decreases to become smaller than 1/k afterwards?

 

[mfb]

Firstly I am right in thinking that "pair wise distinct" means that the speed of all runners are different?

Yes. If two runners have the same speed, then they are never lonely, and the problem gets trivial.

Runners can achieve loneliness independent of other runners (otherwise the problem would be trivial again).

Furthermore, once a runner achieves "loneliness" is it maintained? In other words say a runner is lonely, does it stay lonely even if the distance between it and the other runners decreases to become smaller than 1/k afterwards?

Sure.