Erdős-Rado theorem

Repeated exponentiation for cardinals is denoted ${\exp }_i(\kappa )$, where . It is defined by:

$${\exp }_0(\kappa )=\kappa$$

and

$${\exp }_{i+1}(\kappa )=2^{{\exp }_i(\kappa )}$$

The Erdős-Rado theorem states that:

$${\exp }_i{(\kappa )}^{+}\to {({\kappa }^{+})}_{\kappa }^{i+1}$$

That is, if $f:{[{\exp }_i{(\kappa )}^{+}]}^{i+1}\to \kappa$ then there is a homogeneous set of size ${\kappa }^{+}$.

As special cases, ${(2^{\kappa })}^{+}\to {({\kappa }^{+})}_{\kappa }^2$ and ${(2^{{\mathrm{\aleph }}_0})}^{+}\to {({\mathrm{\aleph }}_1)}_{{\mathrm{\aleph }}_0}^2$.