# 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 $\beta$

 $\kappa\rightarrow(\lambda)^{\alpha}_{\beta}$

states that for any set $X$ of size $\kappa$ and any function $f:[X]^{\alpha}\rightarrow\beta$, there is some $Y\subseteq X$ and some $\gamma\in\beta$ such that $|Y|=\lambda$ and for any $y\in[Y]^{\alpha}$, $f(y)=\gamma$.

As an example, the pigeonhole principle  is the statement that if $n$ is finite and $k then:

 $n\rightarrow 2^{1}_{k}$

That is, if you try to partition $n$ into fewer than $n$ pieces then one piece has more than one element  .

Observe that if

 $\kappa\rightarrow(\lambda)^{\alpha}_{\beta}$

then the same statement holds if:

 $\kappa\nrightarrow(\lambda)^{\alpha}_{\beta}$

References

 Title arrows relation Canonical name ArrowsRelation Date of creation 2013-03-22 17:48:54 Last modified on 2013-03-22 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