Ramsey cardinal

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In mathematics, a Ramsey cardinal (named after Frank P. Ramsey) is a certain kind of large cardinal number.

Formally, a cardinal number κ such that for every function f:[ κ] ^{< ω} → {0, 1} (with [κ] ^{< ω} denoting the set of all finite subsets of κ) there is a set A of cardinality κ that is homogeneous for f (i.e.: for every n, f is constant on the n-tuples from A) is called Ramsey.

The existence of a Ramsey cardinal is sufficient to prove the existence of 0#. In fact, if κ is Ramsey, then every set with rank less than κ has a sharp.

[edit] See also

• Almost Ramsey cardinal
• Ineffably Ramsey cardinal

[edit] References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0444105352.

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3540003843.