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 ${{𝒫}}_n(D)$ be the set of all size-$n$ subsets of $D$ (see Powerset § Subsets of limited cardinality) and let $f:{{𝒫}}_n(D)\to B$ be a function defined in this set. Then $S$ is homogeneous for $D$ if $|f^{″}([S{]}^n)|=1$.^{p. 72p. 1} Ramsey's theorem can be stated as for all functions $f:{\mathbb{\mathbb{N}}}^m\to n$, there is an infinite set $H\subseteq \mathbb{\mathbb{N}}$ which is homogeneous for $f$.^{p. 1}

Partitions of finite subsets

Given a set D, let ${{𝒫}}_{<\omega }(D)$ be the set of all finite subsets of $D$ (see Powerset § Subsets of limited cardinality) and let $f:{{𝒫}}_{<\omega }(D)\to B$ 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 ${{𝒫}}_n(S)$. That is, f is constant on the unordered n-tuples of elements of S.

See also

• Ramsey's theorem
• Ramsey cardinal

External links

• S. Unger, "Introduction to Large Cardinals".