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 $$f:[D]^n\to\lambda$$ if f is constant on size-$$n$$ subsets of S. p. 72 More precisely, given a set D, let $$\mathcal{P}_n(D)$$ be the set of all size-$$n$$ subsets of $$D$$ (see ) and let $$f: \mathcal{P}_n(D) \to B$$ be a function defined in this set. Then $$S$$ is homogeneous for $$D$$ if $$\vert f''([S]^n)\vert=1$$. p. 72 p. 1

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

Partitions of finite subsets
Given a set D, let $$\mathcal{P}_{<\omega}(D)$$ be the set of all finite subsets of $$D$$ (see ) and let $$f: \mathcal{P}_{<\omega}(D) \to B$$ be a function defined in this set. On these conditions, S is homogeneous for f if, for every natural number n, f is constant in the set $$\mathcal{P}_{n}(S)$$. That is, f is constant on the unordered n-tuples of elements of S.