arrows relation
Let ${[X]}^{\alpha}=\{Y\subseteq X\mid Y=\alpha \}$, that is, the set of subsets of $X$ of size $\alpha $. Then given some cardinals $\kappa $, $\lambda $, $\alpha $ and $\beta $
$$\kappa \to {(\lambda )}_{\beta}^{\alpha}$$ 
states that for any set $X$ of size $\kappa $ and any function $f:{[X]}^{\alpha}\to \beta $, there is some $Y\subseteq X$ and some $\gamma \in \beta $ such that $Y=\lambda $ and for any $y\in {[Y]}^{\alpha}$, $f(y)=\gamma $.
In words, if $f$ is a partition^{} of ${[X]}^{\alpha}$ into $\beta $ subsets then $f$ is constant on a subset of size $\lambda $ (a homogeneous^{} subset).
As an example, the pigeonhole principle^{} is the statement that if $n$ is finite and $$ then:
$$n\to {2}_{k}^{1}$$ 
That is, if you try to partition $n$ into fewer than $n$ pieces then one piece has more than one element^{}.
Observe that if
$$\kappa \to {(\lambda )}_{\beta}^{\alpha}$$ 
then the same statement holds if:

•
$\kappa $ is made larger (since the restriction^{} of $f$ to a set of size $\kappa $ can be considered)

•
$\lambda $ is made smaller (since a subset of the homogeneous set will suffice)

•
$\beta $ is made smaller (since any partition into fewer than $\beta $ pieces can be expanded by adding empty sets^{} to the partition)

•
$\alpha $ is made smaller (since a partition $f$ of ${[\kappa ]}^{\gamma}$ where $$ can be extended to a partition ${f}^{\prime}$ of ${[\kappa ]}^{\alpha}$ by ${f}^{\prime}(X)=f({X}_{\gamma})$ where ${X}_{\gamma}$ is the $\gamma $ smallest elements of $X$)
$$\kappa \nrightarrow {(\lambda )}_{\beta}^{\alpha}$$ 
is used to state that the corresponding $\to $ relation^{} is false.
References

•
Jech, T. Set Theory^{}, SpringerVerlag, 2003

•
Just, W. and Weese, M. Topics in Discovering Modern Set Theory, II, American Mathematical Society, 1996
Title  arrows relation 
Canonical name  ArrowsRelation 
Date of creation  20130322 17:48:54 
Last modified on  20130322 17:48:54 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  5 
Author  Henry (455) 
Entry type  Definition 
Classification  msc 05A18 
Classification  msc 03E05 
Related topic  PartitionsLessThanCofinality 
Related topic  ErdosRadoTheorem 
Defines  homogeneous 
Defines  arrows 
Defines  homogeneous set 
Defines  homogeneous subset 