Timely Lonely Runner Conjecture

Ascertain whether for every integer n ≥ 1 there exists a constant N (depending only on n) such that for any set V of n positive speeds, the earliest time t0 at which the origin becomes lonely satisfies t0 ≤ N when times are normalized by the slowest speed.

Background

This conjecture strengthens the Lonely Runner claim by bounding the time to loneliness uniformly in n. It aligns with results that reduce the problem to finitely many bounded-speed instances and has been proven for n ≤ 5.

The problem connects to algorithmic and quantitative aspects of the conjecture, seeking universal temporal bounds independent of particular speed sets.

References

Conjecture 29 (Timely LR Conjecture). For every n ∈ N there is N such that for every n-set V of positive speeds t0≤ N .

— The Lonely Runner Conjecture turns 60  (2409.20160 - Perarnau et al., 2024) in Conjecture 29, Section 10.4 (Time to get lonely)

An effective version of Lemma \ref{lem:reduction} would make Theorems \ref{thm:six} and \ref{thm:five} unconditional statements about all near-tight tuples. The constants in are explicit but enormous; is there a direct argument bounding the speeds of a near-tight $n$-tuple that does not lie on a critical subtorus?

— Odd denominators in the Lonely Runner spectrum for six speeds  (2609.03444 - Cordella, 3 Sep 2026) in Question 3, Section 4.4, “Questions”