Introduces the mixed lonely runner property MLPS_k and exactly characterizes MLPS_2 while deriving Fourier-based summation and integral formulas for unequal thresholds.
The lonely runner with seven runners
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abstract
Suppose $k+1$ runners having nonzero constant speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least $1/(k+1)$ along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners ($k\le 5$). A formulation of the problem is related to the regular chromatic number of distance graphs. We use a new tool developed in this context to solve the first open case of the conjecture with seven runners.
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2026 1verdicts
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Mixed thresholds in the Lonely Runner Conjecture
Introduces the mixed lonely runner property MLPS_k and exactly characterizes MLPS_2 while deriving Fourier-based summation and integral formulas for unequal thresholds.