Graphs and Ultrapowers

Abstract

<p> Graphs are defined as a special kind of relational system and an analogue of Birkhoff's Representation Theorem for Universal Algebras is proved. The notion of ultrapower, a specialization of the ultraproducts introduced into Mathematical Logic by Tarski, Scott and others, is demonstrated to provide a unifying framework within which various problems of graph theory and infinite combinatorial mathematics can be formulated and solved. Thus, theorems extending to the infinite case results of N.G.de Bruijn and P.Erdős in graph colouring, and of P. and M. Hall in combinatorial set theory are proved via the method of ultrapowers. Finally, the problem of embedding graphs in certain topological spaces is taken up, and a characterization of infinite connected planar graphs is derived (see Introduction). </p>