 Partition relation

The notation $\lambda \to (\mu {)}_{\beta }^{\alpha }$ says that every $\beta$-coloring of the $\alpha$-element subsets of $\lambda$ has a homogeneous subset of order type $\mu$. At infinite arity one specifies whether the domain subsets have a fixed cardinality or a fixed order type; the usual finite-arity notation has no such ambiguity.