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The Pigeonhole Principle

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Extremal Combinatorics

Part of the book series: Texts in Theoretical Computer Science. An EATCS Series ((TTCS))

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Abstract

The pigeonhole principle (also known as Dirichlet’s principle) states the “obvious” fact that n+1 pigeons cannot sit in n holes so that every pigeon is alone in its hole. More generally, the pigeonhole principle states the following:

If a set consisting of at least rs+1 objects is partitioned into r classes, then some class receives at least s+1 objects.

Its truth is easy to verify: if every class receives at most s objects, then a total of at most rs objects have been distributed. To see that the result is best possible, observe that a set with at most rs points can be divided into r groups with at most s points in each group; hence none of the groups contains s+1 points.

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Correspondence to Stasys Jukna .

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© 2011 Springer-Verlag Berlin Heidelberg

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Jukna, S. (2011). The Pigeonhole Principle. In: Extremal Combinatorics. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17364-6_4

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  • DOI: https://doi.org/10.1007/978-3-642-17364-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-17363-9

  • Online ISBN: 978-3-642-17364-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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