Ramsey goodness and generalized stars

Let G and H be fixed graphs with s ( G ) = s (the minimum number of vertices in a color class over all proper vertex-colorings of G with χ ( G ) colors). It is shown that r ( K 1 + G , K 1 + n H ) ≤ k ( h n + s − 1 ) + 1 for large n , where χ ( G ) = k ≥ 2 . In particular, if s is odd or s is even and h n is odd, then r ( K 1 + K k ( s ) , K 1 + n H ) = k ( h n + s − 1 ) + 1 , where K k ( s ) is a complete k -partite graph with s vertices in each part, implying that K 1 + n H is not ( K 1 + K k ( s ) ) -good. Moreover, r ( K 1 + s K 2 , K 1 + n H ) = 2 h n + 1 for large n .