Ramsey Properties of Families of Graphs

For a graph F and natural numbers a1,…,ar, let F→(a1,…,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)→(b1,…,bs) if for every F for which F→(a1,…,ar) we have also F→(b1,…,bs). In this note, we show that a trivial sufficient condition for the relation (a1,…,ar)→(b1,…,bs) is necessary as well.