Existence of Configurations

Caminha Muniz Neto, Antonio

doi:10.1007/978-3-319-77977-5_4

Existence of Configurations

Antonio Caminha Muniz Neto³

Chapter
First Online: 18 April 2018

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Part of the book series: Problem Books in Mathematics ((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.

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Notes

1.
After the German mathematician of the nineteenth century Gustav L. Dirichlet. Actually, we shall have to wait until Sect. 7.2 (cf. Lemma 7.7) to appreciate the original problem that motivated Dirichlet to introduce the pigeonhole principle.
2.
Former name of the Russian city of Saint Petersburg.
3.
Let φ(n) denote expression (2.7), deduced at Problem 6, page 40, and let \(a\in \mathbb Z\) be relatively prime with n. In Sect. 10.2, we shall prove a theorem of L. Euler which assures that a ^φ(n) always leave remainder 1 upon division by n (in the language of Sect. 2.3,a ^φ(n) ≡ 1(mod n)).
4.
James Sylvester , English mathematician, and Tibor Gallai , Hungarian mathematician, both of the twentieth century.
5.
Nicolaas de Bruijn , Dutch mathematician of the twentieth century.
6.
After Leon Mirsky , Russian mathematician of the twentieth century.
7.
After Robert Dilworth , American mathematician of the twentieth century.
8.
Formally, an algorithm is a finite sequence of well defined operations that, once performed on some (more or less) arbitrary set of data (called the input of the algorithm), furnish a definite result, called the outcome or output of the algorithm. On the other hand, each such performance of the algorithm is generally referred to as an iteration of it. We shall encounter algorithms several times along these notes.

References

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L. Mirsky, A dual of Dilworth’s decomposition theorem. Am. Math. Monthly. 78, 876–877 (1971)
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Universidade Federal do Ceará, Fortaleza, Ceará, Brazil
Antonio Caminha Muniz Neto

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Caminha Muniz Neto, A. (2018). Existence of Configurations. In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_4

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DOI: https://doi.org/10.1007/978-3-319-77977-5_4
Published: 18 April 2018
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77976-8
Online ISBN: 978-3-319-77977-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)

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