On the Chromatic Numbers of Infinite Graphs

In this paper, we survey some of the results and open questions regarding colorings of infinite graphs. We first introduce terminology to describe finite and infinite colorings, then offer a proof of the De Bruijn-Erdős Theorem, one of the primary tool for studying the chromatic number of infinite graphs. We show how the De Bruijn-Erdős Theorem allows us to extend results of finite graphs of arbitrarily high chromatic number to infinite graphs which are not finitely-colorable, while avoiding certain subgraphs. We use Mycielski’s construction as an example of this phenomenon. Then we examine the natural extension of the Four-Color Theorem into three dimensions, where the best bound is that all three dimensional maps are countably colorable. Finally, we discuss two open questions regarding the coloring of infinite graphs: whether all cardinals smaller than the chromatic number of a graph appear as the chromatic number of a subgraph, and how many colors are required to properly color the plane.