• John StillwellEmail author
Part of the Undergraduate Texts in Mathematics book series (UTM)


In this final chapter we look at another field that came to maturity in the 20th century: combinatorics. Like number theory before the 19th century, combinatorics before the 20th century was thought to be an elementary topic without much unity or depth. We now realize that, like number theory, combinatorics is infinitely deep and linked to all parts of mathematics. Here we emphasize the parts that link nicely to topics from earlier chapters, but without completely sacrificing the distinctive features of the subject. Combinatorics is often called “finite mathematics” because it studies finite objects. But there are infinitely many finite objects, and it is sometimes convenient to reason about all members of an infinite collection at once. In fact, combinatorics pioneered this idea with the use of generating functions (already seen in Section 10.6). Other important infinite principles in combinatorics are the infinite pigeonhole principle and the Kőnig infinity lemma. We illustrate these first by some classical proofs in number theory and analysis, then in the 20thcentury fields of graph theory and Ramsey theory. Ramsey theory leads us to a proof of the Paris–Harrington theorem, mentioned in Section 24.8 as a theorem that cannot be proved in the strictly finite reasoning of PA. Infinite reasoning is likewise essential for graph theory. The field had its origins in topology, and it is still relevant there, but it has expanded extraordinarily far in other directions. Graph theory today is exploring the boundaries of finite provability first exposed by Gödel’s incompleteness theorem.


Petersen Graph Peano Arithmetic Pigeonhole Principle Blue Edge Prime Number Theorem 
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  1. Ackermann, W. F. (1937). Der Widerspruchsfreiheit der allgemeine Mengenlehre. Math. Ann. 112, 305–315.CrossRefMathSciNetGoogle Scholar
  2. Appel, K. and W. Haken (1976). Every planar map is four colorable. Bull. Amer. Math. Soc. 82, 711–712.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Diestel, R. (2005). Graph Theory (Third ed.). Berlin: Springer-Verlag.zbMATHGoogle Scholar
  4. Graham, R. L., B. L. Rothschild, and J. H. Spencer (1990). Ramsey Theory (Second ed.). New York: John Wiley & Sons Inc.zbMATHGoogle Scholar
  5. Hadamard, J. (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bull. Soc. Math. France 24, 199–220.zbMATHMathSciNetGoogle Scholar
  6. Jordan, C. (1887). Cours de Analyse de l’École Polytechnique. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  7. Kruskal, J. B. (1960). Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Trans. Amer. Math. Soc. 95, 210–225.zbMATHMathSciNetGoogle Scholar
  8. Kuratowski, K. (1930). Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15, 271–283.zbMATHGoogle Scholar
  9. Ramsey, F. P. (1929). On a problem of formal logic. Proc. Lond. Math. Soc. 30, 291–310.Google Scholar
  10. Schechter, B. (1998). My Brain Is Open. New York: Simon & Schuster.Google Scholar
  11. Soifer, A. (2009). The Mathematical Coloring Book. New York: Springer.zbMATHGoogle Scholar
  12. Sperner, E. (1928). Neuer Beweis für die Invarianz der Dimensionzahl und des Gebietes. Abh. Math. Sem. Univ. Hamburg 6, 265–272.zbMATHCrossRefGoogle Scholar
  13. Veblen, O. (1905). Theory of plane curves in nonmetrical analysis situs. Trans. Amer. Math. Soc. 6, 83–98.zbMATHMathSciNetGoogle Scholar
  14. Wagner, K. W. (1936). Bemerkungen zum Vierfarbenproblem. Jahresber. Deutsch. Math.-Ver. 46, 26–32.Google Scholar
  15. Wagner, K. W. (1937). Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114, 570–590.CrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

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