<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> / Thesis / Doctor of Philosophy (PhD)

The results from this dissertation are an exact computation of ultrapowers by measures on cardinals $\aleph\sb{n},\ n\in w$, in $L(\IR$), and a proof that ordinals in $L(\IR$) below $\delta\sbsp{5}{1}$ represented by descriptions and the identity function with respect to sequences of measures are cardinals. An introduction to the subject with the basic definitions and well known facts is presented in chapter I. In chapter II, we define a class of measures on the $\aleph\sb{n},\ n\in\omega$, in $L(\IR$) and derive a formula for an exact computation of the ultrapowers of cardinals by these measures. In chapter III, we give the definitions of descriptions and the lowering operator. Then we prove that ordinals represented by descriptions and the identity function are cardinals. This result combined with the fact that every cardinal $<\delta\sbsp{5}{1}$ in $L(\IR$) is represented by a description (J1), gives a characterization of cardinals in $L(\IR$) below $\delta\sbsp{5}{1}. Concrete examples of formal computations are shown in chapter IV.