Ramsey families which exclude a graph

For graphsA andB the relationA?(B)_r¹ means that for everyr-coloring of the vertices ofA there is a monochromatic copy ofB inA. Forb (G) is the family of graphs which do not embedG. A familyFof graphs is Ramsey if for all graphsB?Fthere is a graphA?Fsuch thatA?(B)_r¹. The only graphsG for which it is not known whether Forb (G) is Ramsey are graphs which have a cutpoint adjacent to every other vertex except one. In this paper we prove for a large subclass of those graphsG, that Forb (G) does not have the Ramsey property.