Partition relation ($\lambda \to (\mu {)}_{\beta }^{\alpha }$)

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.

The relation $\lambda \to (\lambda {)}_{\beta }^{<\omega }$ requires a subset of size $\lambda$ homogeneous at each finite arity. The constant color may depend on the arity.

If inaccessible $\lambda$ has the finite-subset partition property for every color number below $\lambda$, forcing of size below $\lambda$ preserves its two-color version. Color ground-model finite sets by their full forcing decision patterns; strong-limit cardinal arithmetic bounds the number of patterns.

A partition relation whose colored subsets are infinite. For countably infinite arity, choosing representatives modulo finite symmetric difference and coloring its parity shows $\kappa \nrightarrow (\omega {)}_2^{\omega }$ for every infinite cardinal $\kappa$.