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 .[2]p. 1
Partitions of finite subsets
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]
See also
References
- ^ a b F. Drake, Set Theory: An Introduction to Large Cardinals (1974).
- ^ a b Cody, Brent (2020). "A Refinement of the Ramsey Hierarchy Via Indescribability". The Journal of Symbolic Logic. 85 (2): 773–808. arXiv:1907.13540. doi:10.1017/jsl.2019.94.
External links
- S. Unger, "Introduction to Large Cardinals".