This book evolved from several courses in combinatorics and graph theory given at Appalachian State University and UCLA. Chapter 1 focuses on finite graph theory, including trees, planarity, coloring, matchings, and Ramsey theory. Chapter 2 studies combinatorics, including the principle of inclusion and exclusion, generating functions, recurrence relations, Pólya theory, the stable marriage problem, and several important classes of numbers. Chapter 3 presents infinite pigeonhole principles, König's lemma, and Ramsey's theorem, and discusses their connections to axiomatic set theory. The text is written in an enthusiastic and lively style. It includes results and problems that cross subdisciplines, emphasizing relationships between different areas of mathematics. In addition, recent results appear in the text, illustrating the fact that mathematics is a living discipline. The text is primarily directed toward upper-division undergraduate students, but lower-division undergraduates with a penchant for proof and graduate students seeking an introduction to these subjects will also find much of interest.
Table of contents:
Graph Theory: Introductory Concepts.- Trees.- Planarity.- Colorings.- Matchings.- Ramsey Theory.- References; Combinatorics: Three Basic Problems.- Binomial Coefficients.- The Principle of Inclusion and Exclusion.- Generating Functions.- Polya's Theory of Counting.- More Numbers.- Stable Marriage.- References; Infinite Combinatorics and Graph Theory: Pigeons and Trees.- Ramsey Revisited.- ZFC.- The Return of der Koenig.- Ordinals, Cardinals, and Many Pigeons.- Incompleteness and Coardinals.- Weakly Compact Cardinals.- Finite Combinatorics with Infinite Consequences.- Points of Departure.- References.
Table of contents:
Graph Theory: Introductory Concepts.- Trees.- Planarity.- Colorings.- Matchings.- Ramsey Theory.- References; Combinatorics: Three Basic Problems.- Binomial Coefficients.- The Principle of Inclusion and Exclusion.- Generating Functions.- Polya's Theory of Counting.- More Numbers.- Stable Marriage.- References; Infinite Combinatorics and Graph Theory: Pigeons and Trees.- Ramsey Revisited.- ZFC.- The Return of der Koenig.- Ordinals, Cardinals, and Many Pigeons.- Incompleteness and Coardinals.- Weakly Compact Cardinals.- Finite Combinatorics with Infinite Consequences.- Points of Departure.- References.
From the reviews: SIAM REVIEW "The narrative and proofs are well written, and the authors are given to frequent uses of humor. Students should find this book as easy to read as any other good-quality text written with them in mind. Each of the three chapters concludes with several paragraphs describing an excellent selection of more advanced texts or papers to consider for further study".