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In set theory and in the context of a large cardinal property, a subset, S, of D is homogeneous for a function ${\displaystyle f:^{n}\to \lambda }$ if f is constant on size-${\displaystyle n}$ subsets of S. 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 ${\displaystyle \vert f''(^{n})\vert =1}$.

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 ${\displaystyle f}$.

Partitions of finite subsets

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

See also

• Ramsey's theorem
• Ramsey cardinal

References

1. ^ F. Drake, Set Theory: An Introduction to Large Cardinals (1974).
2. ^ 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".

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