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{\bf Tom Bohman, Ron Holzman and Dan Kleitman}
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{\bf Six Lonely Runners}
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For $x$ real, let $ \{x\}$  be the fractional part of $x$
(i.e. $ \{x\} = x - \lfloor x \rfloor $).  In this paper we
prove the $k=5$ case of the following conjecture
(the lonely runner conjecture):  for any $k$ positive reals
$ v_1, \dots , v_k $ there exists a real number $t$
such that $ 1/(k+1) \le \{v_it \} \le k/(k+1) $ for
$ i= 1, \dots, k$.


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