Ramsey Numbers Involving Graphs with Long Suspended Paths

Let G be a graph with chromatic number χ and with t being the minimum number of points in any color class of any point‐coloring of G with χ colors. Let H be any connected graph and let H n be a graph on n points which is homeomorphic to H . It is proved that if n is large enough, the Ramsey number r ( G , H n ) satisfies r ( G , H n ) = (χ − l)( n −l) + t . It is also shown that for some G , no such result holds when H n is a star with n points.