Ramsey properties of families of graphs

Abstract

For a graph F and natural numbers al, ⋯, ar, let F → (al,...,ar) denote the property that for each coloring of the edges of F with r colors, there exists i such that some copy of the complete graph Kai is colored with the ith color. Furthermore, we write (a1,...,ar) → (bl,...,bs) if for every F for which F → (al,...,ar) we have also F → (bl,...,bs). In this note, we show that a trivial sufficient condition for the relation (al,...,ar) → (bl,...,bs) is necessary as well. © 2002 Elsevier Science (USA).