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Statement

Suppose

k \geq 2

runners run forever on a unit-circumference circular track at distinct constant nonzero speeds

v_1, \ldots, v_k

, all starting at the origin at time

t = 0

. Conjecture: for every runner

i

there exists a time

t > 0

at which the (arc-length, circular) distance from runner

i

to every other runner is at least

1/k

. Equivalently: for any

k - 1

positive reals

w_1, \ldots, w_{k-1}

(the relative speeds), there exists

t \geq 0

with

\| t w_j \| \geq 1/k

for all

j

, where

\|\cdot\|

is the distance to the nearest integer.

Current frontier

Proved for all k ≤ 10 (k ≤ 7 by 2008; k = 8 by Rosenfeld 2025, arXiv:2509.14111; k = 9, 10 by Trakulthongchai 2025, arXiv:2511.22427, and independently Rosenfeld for k = 9 in arXiv:2512.01912). First open case is k = 11.

When this counts as solved

A full solve is a rigorous proof of the conjecture for all

k \geq 2

, or an explicit

k

and speed tuple

(v_1, \ldots, v_k)

where some runner is never lonely. Settling the next open case (

k = 11

, or any larger

k

not already covered by the published 2025 work) also counts as a breakthrough — but the proof has to handle the full integer-speed parameter space for that

k

. Improved view-obstruction bounds that don't settle a new

k

, re-proofs of already-settled cases, or tightened loneliness bounds

1/k + o(1)

don't close it.

Classification

open-completion