For distinct positive integers v₁,…,\(v_{n}\) and any d ∈ {1,…,n}, there exists t such that ||t·\(v_{i}\)|| ≥ 1/(n+1) for all i (distances on the unit circle), i.e. runner d is 'lonely'. Equivalently: max over t of \(min_{i}\) ||t·\(v_{i}\)|| ≥ 1/(n+1).
Wills posed it in 1968 as a view-obstruction question; the running metaphor came via Goddyn. Proved for n ≤ 7 by hand and computation; Tao (2018) proved it for 'almost all' speed tuples. Its truth would imply density results for chromatic numbers of distance graphs. Utterly elementary to state — and untouched by every technique thrown at it.
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