Weakly compact cardinal

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In mathematics, a weakly compact cardinal is a certain kind of cardinal number; weakly compact cardinals are large cardinals, meaning that their existence can neither be proven nor disproven from the standard axioms of set theory.

Formally, a cardinal κ is weakly compact iff for every function f: κ ² → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, a subset S of κ is homogeneous for f iff either all of S×S maps to 0 or all of it maps to 1.

Theorem

The following are equivalent for any uncountable cardinal κ:

1. κ is weakly compact.
2. for every λ<κ, natural number n ≥ 2, and function f: κⁿ → λ, there is a set of cardinality κ that is homogeneous for f.
3. κ is inaccessible and every tree of height κ either has a path or a level of cardinality at least κ.
4. Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
5. κ is <math>\Pi^1_1</math> indescribable.
6. For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
7. κ is κ-unfoldable.