The partition relation
$$ \kappa \to (\alpha)^m_\lambda $$
says that for any $f:[\kappa]^m \to \lambda$, there is a $X\subseteq \kappa$ such that $f$ is constant on $[X]^m$ and the order type of $X$ is $\alpha$.
(My introduction to this notation was from Drake's book on large cardinals, but I believe that it comes from Erdős and Rado.)
My question: how do you pronounce this relation when reading it out loud (if there is a standard way)?
Read it as "$\kappa$ arrows $\alpha$-$m$-$\lambda$", or some small variant of this, such as "$\kappa$ arrows $\alpha$ super $m$ sub $\lambda$". Nothing too enlightening. That this is indeed the suggested reading is indicated in section 8.2 (pg. 53) of the Erdős-Hajnal-Máté-Rado book, Combinatorial set theory: Partition relations for cardinals.
(The notation indeed comes from a paper by Erdős and Rado, but it was designed by Rado in the early 50s. My understanding is that Erdős would complain about it from time to time, in spite of its usefulness.)