Homogeneous (large cardinal property) explained

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\toλ} if f is constant on size-n subsets of S.^{[1] p. 72} More precisely, given a set D, let l{P} _{n(D)} be the set of all size-n subsets of D (see) and let f: l{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[2] p. 1}

Ramsey's theorem can be stated as for all functions

f: N ^{m\to} n , there is an infinite set H\subseteq N which is homogeneous for f .^{p. 1}

Partitions of finite subsets

Given a set D, let

l{P} _{<\omega} (D) be the set of all finite subsets of D (see) and let f: l{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 l{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".

Notes and References

1. F. Drake, Set Theory: An Introduction to Large Cardinals (1974).
2. 1907.13540 . 10.1017/jsl.2019.94 . A Refinement of the Ramsey Hierarchy Via Indescribability . 2020 . Cody . Brent . The Journal of Symbolic Logic . 85 . 2 . 773–808 .