In set theory and in the context of a large cardinal property, a subset, S, of D is homogeneous for a function if f is constant on size- subsets of S. [1]p. 72 More precisely, given a set D, let be the set of all size- subsets of (see Powerset § Subsets of limited cardinality) and let be a function defined in this set. Then is homogeneous for if . [1]p. 72 [2]p. 1
Ramsey's theorem can be stated as for all functions , there is an infinite set which is homogeneous for \(f\). [2]p. 1
Given a set D, let be the set of all finite subsets of (see Powerset § Subsets of limited cardinality) and let 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 . That is, f is constant on the unordered n-tuples of elements of S.[ citation needed]