Lower bounds for the maximum number of runners that cause loneliness, and its application to Isolation
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We consider (n+1) runners with given constant unique integer speeds running along the circumference of a circle whose circumferential length is one, and all runners starting from the same point. We define and give lower bounds to a first problem PMAX of finding, for every runner r, the maximum number of runners that can be simultaneously separated from runner r by a distance of atleast d. For d=1/(2^(floor(lg(n)))), a lower bound for PMAX is ( n - ((n-1)/floor(lg(n))) ), which makes the fraction of simultaneously separated runners tend to 1 as n tends to infinity. Next, we define and give upper bounds to a second problem ISOLATE of finding, for every runner r, the minimum number of steps needed to isolate r, assuming that the runners that can be simultaneously separated from r by atleast d, are removed at each step. For d=1/(2^(floor(lg(n)))), an upper bound for ISOLATE is ( lg(n - 1)/lg(floor(lg(n))) ).
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