Math 2.0

Combinatorics Theorem

Infinite pigeonhole principle

Builds on Pigeonhole principle

Statement

Let AA be an infinite set, let BB be a finite set, and let f:A→Bf : A \to B be a function. Then there is an element b∈Bb \in B such that the fibre f−1(b)f^{-1}(b) is infinite.

Remarks

The proof is short. If each of the finitely many fibres were finite, then AA, their union, would be finite. Example: in any infinite sequence of integers, infinitely many terms have the same remainder modulo 1010. The statement is the starting point of the infinite form of Ramsey's theorem.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Cardinality of a finite set, definition, no evidence levelCardinality of a fini…definitionPigeonhole principle, statement, no evidence levelPigeonhole principlestatementInfinite pigeonhole principle, statement, no evidence levelInfinite pigeonhole p…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cardinality of a finite set definition
  2. Pigeonhole principle statement
  3. Infinite pigeonhole principle statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Pigeonhole_principle, Infinite sets
About this page
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Kindstatement, theorem
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MSC05A05
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UsesPigeonhole principle

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