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 $y\in [Y]^\alpha$ $f(y)=\gamma$

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 $k<n$ 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:

• $\kappa$ 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 $\gamma<\alpha$ 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 $X$

$$\kappa\nrightarrow(\lambda)^\alpha_\beta$$

is used to state that the corresponding $\rightarrow$ 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