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Let
![$ [X]^\alpha=\{Y\subseteq X\mid \vert Y\vert=\alpha\}$](http://images.planetmath.org:8080/cache/objects/10278/l2h/img1.png "$ [X]^\alpha=\{Y\subseteq X\mid \vert Y\vert=\alpha\}$") , that is, the set of subsets of  of size  . Then given some cardinals  ,  ,  and 
states that for any set  of size  and any function
![$ f:[X]^\alpha\rightarrow\beta$](http://images.planetmath.org:8080/cache/objects/10278/l2h/img11.png "$ f:[X]^\alpha\rightarrow\beta$") , there is some
 and some
 such that
 and for any
![$ y\in [Y]^\alpha$](http://images.planetmath.org:8080/cache/objects/10278/l2h/img15.png "$ y\in [Y]^\alpha$") ,
 .
In words, if  is a partition of
![$ [X]^\alpha$](http://images.planetmath.org:8080/cache/objects/10278/l2h/img18.png "$ [X]^\alpha$") into  subsets then  is constant on a subset of size  (a homogeneous subset).
As an example, the pigeonhole principle is the statement that if  is finite and  then:
That is, if you try to partition  into fewer than  pieces then one piece has more than one element.
Observe that if
then the same statement holds if:
 is made larger (since the restriction of  to a set of size  can be considered) is made smaller (since a subset of the homogeneous set will suffice) is made smaller (since any partition into fewer than  pieces can be expanded by adding empty sets to the partition) is made smaller (since a partition  of
![$ [\kappa]^\gamma$](http://images.planetmath.org:8080/cache/objects/10278/l2h/img36.png "$ [\kappa]^\gamma$") where
 can be extended to a partition  of
![$ [\kappa]^\alpha$](http://images.planetmath.org:8080/cache/objects/10278/l2h/img39.png "$ [\kappa]^\alpha$") by
 where  is the  smallest elements of  )
is used to state that the corresponding
 relation is false.
References
- Jech, T. Set Theory, Springer-Verlag, 2003
- Just, W. and Weese, M. Topics in Discovering Modern Set Theory, II, American Mathematical Society, 1996
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