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Abstract  LetH  1   ,  ...  ,   H  kbe graphs. The multicolor Ramsey number    r  (   H  1   ,  ...  ,   H  k   )    is the minimum integer  r  such that in every edge‐coloring of    K  r    by  k  colors, there is a monochromatic copy of    H  i    in color  i  for some    1  ≤  i  ≤  k   . In this paper, we investigate the multicolor Ramsey number    r  (   K   2  ,  t    ,  ...  ,   K   2  ,  t    ,   K  m   )   , determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of  t  and  m . Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and Rödl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result isc  1m  2   tlog  4    (  m  t  )≤  r   (   K   2  ,  t    ,   K   2  ,  t    ,   K  m   )   ≤   c  2m  2   tlog  2   mfor any  t  and  m , where  c  1  and  c  2  are absolute constants.

Multicolor Ramsey Numbers For Complete Bipartite Versus Complete Graphs