pigeonhole principle

Etymology
From the commonly used expository example that if n+1 pigeons are placed in n pigeonholes, at least one pigeonhole must contain two (or more) pigeons.

Noun

 * 1)  The theorem which states that any partition of a finite set of n elements into m (< n) subsets (allowing empty subsets) must include a subset with two or more elements; any of certain reformulations concerning the partition of infinite sets where the cardinality of the unpartitioned set exceeds that of the partition (so there is no one-to-one correspondence).
 * 2) * 2009, John Harris, Jeffry L. Hirst, Michael Mossinghoff, Combinatorics and Graph Theory, Springer, page 313,
 * Of course our list of pigeonhole principles is not all inclusive. For example, more set theoretic pigeonhole principles are given in [72].
 * Corollary 3.31 (Ultimate Pigeonhole Principle). The following are equivalent:
 * 1. κ is a .
 * 2. If we put κ pigeons into λ < κ pigeonholes, then some pigeonhole must contain κ pigeons.
 * 1) * 2012,, , (editors), Handbook of the History of Logic: Volume 6: Sets and Extensions in the Twentieth Century, Elevier (North-Holland), page 325,
 * As we turn to look at various pigeonhole principles and how they are used to prove partition theorems, particularly for pairs, we keep in mind the slogan that is embedded in the Motzkin quote: complete disorder is impossible.
 * As we turn to look at various pigeonhole principles and how they are used to prove partition theorems, particularly for pairs, we keep in mind the slogan that is embedded in the Motzkin quote: complete disorder is impossible.

Usage notes
An alternative formulation is that the codomain of an injective function on finite sets cannot be smaller than its domain. With this formulation, no restatement is necessary when infinite sets are considered.

The plural, strictly speaking, refers to formulations of the theorem.

Translations

 * Finnish: kyyhkyslakkaperiaate
 * Polish: zasada szufladkowa
 * Slovak: Dirichletov princíp