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

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

and

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

The Erdos-Rado theorem states that:

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

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

As special cases, $(2^\kappa)^+\rightarrow(\kappa^+)^2_\kappa$ and $(2^{\aleph_0})^+\rightarrow(\aleph_1)^2_{\aleph_0}$ .