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

 $\operatorname{exp}_{0}(\kappa)=\kappa$

and

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

 $\operatorname{exp}_{i}(\kappa)^{+}\rightarrow(\kappa^{+})^{i+1}_{\kappa}$
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 $(2^{\aleph_{0}})^{+}\rightarrow(\aleph_{1})^{2}_{\aleph_{0}}$.