On cycle - Complete graph ramsey numbers

A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K„), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices . Then, another cycle-complete graph Ramsey number is studied, namely r( :C,,,, K„) the smallest order for a graph which forces it to contain either a cycle of order 1 for some I satisfying 3 :1 !~ ,m or a set of n independent vertices . We obtain the exact value of r( :C,,,, K„) for all m > n and an upper bound which applies when m is large in comparison with log n .