Reverse Mathematics and the Coloring Number of Graphs

We use methods of Reverse Mathematics to analyze the proof theoretic strength of certain graph theoretic theorems involving the notion of coloring number. Classically , the coloring number of a graph G = (V, E) is the least cardinal κ such that there is a well ordering of V such that below any vertex in V , there are fewer than κ many vertices connected to it by E. A theorem which we will study in depth, due to Komjáth and Milner, states that if a graph is the union of n forests, then the coloring number of the graph is at most 2n. In particular, we look at the case when n = 1. In doing the above, it is necessary for us to formulate various different Reverse Mathematics definitions of coloring number; we also analyze the relationships between these definitions. 2009 ii ACKNOWLEDGEMENTS I would like to thank my advisors Reed Solomon and Joe Miller. I most certainly would not have made it this far without all of their guidance and support. Despite having three other Ph.D. students, two of which are also graduating this semester, for the past year Reed has always taken the time out of his busy schedule to help me if I had any sort of question or issue; he is indeed an outstanding advisor and teacher. For the past three years, Joe has been an excellent advisor and teacher as well. Even after moving to Wisconsin, Joe has continued meeting with me via Skype every week; he has also given me invaluable guidance throughout the job application process. I could not have asked for better advisors than Reed and Joe. I would also like to thank Manny Lerman, who is so well-respected in the field of Computability Theory and the reason I wanted to come to UConn for graduate school, for being on my advisory committee. I would like to thank Reed Solomon, Joe Miller and Tom Defranco for their wonderful recommendation letters; a college dean, while in a job interview meeting with me, actually made a comment as to how great they were. I would like to thank my graduate school roommates and friends Mike Higdon, Tyler Markkanen, Russell Prime and Bob Wooster for putting up with me for the past few years. I would also like to thank my past and current office-mates iii Oscar Levin (Oscar say, " proof done! …