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We consider (n+1) runners with given constant unique integer speeds runningalong the circumference of a circle whose circumferential length is one, andall runners starting from the same point. We define and give lower bounds to afirst problem PMAX of finding, for every runner r, the maximum number ofrunners that can be simultaneously separated from runner r by a distance ofatleast 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 separatedrunners tend to 1 as n tends to infinity. Next, we define and give upper boundsto a second problem ISOLATE of finding, for every runner r, the minimum numberof steps needed to isolate r, assuming that the runners that can besimultaneously separated from r by atleast d, are removed at each step. Ford=1/(2^(floor(lg(n)))), an upper bound for ISOLATE is ( lg(n -1)/lg(floor(lg(n))) ).

Lower bounds for the maximum number of runners that cause loneliness, and its application to Isolation