For graphsA andB the relationA→(B) r 1 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) r 1 . 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.
Mathematics Subject Classification (1991)05 C 55
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- P. Erdős andA. Hajnal: On chromatic number of graphs and set systems,Acta. Math. Acad. Sci. Hung. 17 (1966), 61–99.Google Scholar
- J. Nešertřil andV. Rödl: Partitions of vertices,Comment. Math. Univ. Carolina. 17, (1976), 85–95.Google Scholar
- V. Rödl andN. Sauer: The Ramsey property for families of graphs which exclude a given graph,Canadian Journal of Mathematics 44 (5), (1992) 1050–1060.Google Scholar