A family of countable homogeneous graphs

Abstract

Let K be the class of all countable graphs and let Kp be the class of all members of K which have no complete subgraphs of cardinality p. R. Rado has constructed a graph U which is universal for. In this paper U is shown to be homogeneous, in the sense of Fraissè. Also a simple construction is given of a graph Gv which is homogeneous and universal for Kp (for each p ≧ 3) and the structure of these graphs is investigated. It is shown that if H is an infinite member of Kp then H can be embedded in Gp in such a way that every automorphism of H extends uniquely to an automorphism of Gp. A similar result holds for U. Also, U and G3 have single-orbit automorphisms, while if p > 3, then Gp has no such automorphism. Finally, a result concerning vertex colorings of the graphs Gp is proved and used to give a new proof of a Theorem of Folkman on vertex colorings of finite graphs. © 1971 Pacific Journal of Mathematics.