Existence of Configurations

  • Antonio Caminha Muniz Neto
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

Generally speaking, Combinatorics deals with two different kinds of problems: those in which we want to count the number of distinct ways of making a certain choice and those in which we want to make sure that some configuration does appear. In order to solve a problem of the first kind above, we employ, among others, the counting techniques discussed in the previous chapters. For the second kind of problem, up to this moment we have not developed any idea that could be systematically used. It is our purpose in this chapter to remedy this state of things. To this end, we start by discussing the famous pigeonhole’s principle of Dirichlet, along with several interesting examples. Then we move on to some applications of the principle of mathematical induction to the existence of configurations. The chapter continues with the study of partial order relations, exploring Mirsky’s theorem on the relation between chains and anti-chains. We close the chapter by explaining how, in some situations, the search for an adequate invariant or a semi-invariant can give the final outcome of certain seemingly random algorithms.

References

  1. 8.
    A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)CrossRefGoogle Scholar
  2. 14.
    R. Dilworth, A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)MathSciNetCrossRefGoogle Scholar
  3. 15.
    P. Erdös, E. Szekeres, A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)MathSciNetMATHGoogle Scholar
  4. 19.
    F. Galvin, A proof of Dilworth’s chain decomposition theorem. Am. Math. Monthly 101, 352–353 (1994)MathSciNetCrossRefGoogle Scholar
  5. 30.
    L. Mirsky, A dual of Dilworth’s decomposition theorem. Am. Math. Monthly. 78, 876–877 (1971)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Antonio Caminha Muniz Neto
    • 1
  1. 1.Universidade Federal do CearáFortalezaBrazil

Personalised recommendations