Homogeneous (large cardinal property)

In set theory and in the context of a large cardinal property, a subset, S, of D is homogeneous for a function ${\displaystyle f:[D]^{n}\to \lambda }$ if f is constant on size-${\displaystyle n}$ subsets of S.^{[1]p. 72} More precisely, given a set D, let ${\displaystyle {\mathcal {P}}_{n}(D)}$ be the set of all size-${\displaystyle n}$ subsets of ${\displaystyle D}$ (see Powerset § Subsets of limited cardinality) and let ${\displaystyle f:{\mathcal {P}}_{n}(D)\to B}$ be a function defined in this set. Then ${\displaystyle S}$ is homogeneous for ${\displaystyle D}$ if ${\displaystyle \vert f''([S]^{n})\vert =1}$.^{[1]p. 72[2]p. 1}

Ramsey's theorem can be stated as for all functions ${\displaystyle f:\mathbb {N} ^{m}\to n}$, there is an infinite set ${\displaystyle H\subseteq \mathbb {N} }$ which is homogeneous for ${\displaystyle f}$.^{[2]p. 1}