In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory.
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: [κ] ^{2} → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] ^{2} means the set of 2element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of [S]^{2} maps to 0 or all of it maps to 1.
The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.
Some authors use a weaker definition of weakly compact cardinals, such as one of the conditions below with the condition of inaccessibility dropped.
Equivalent formulations
The following are equivalent for any uncountable cardinal κ:

κ is weakly compact.

for every λ<κ, natural number n ≥ 2, and function f: [κ]^{n} → λ, there is a set of cardinality κ that is homogeneous for f. (Drake 1974, chapter 7 theorem 3.5)

κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ.

Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.

κ is \Pi^1_1indescribable.

κ has the extension property. In other words, for all U ⊂ V_{κ} there exists a transitive set X with κ ∈ X, and a subset S ⊂ X, such that (V_{κ}, ∈, U) is an elementary substructure of (X, ∈, S). Here, U and S are regarded as unary predicates.

For every set S of cardinality κ of subsets of κ, there is a nontrivial κcomplete filter that decides S.

κ is κunfoldable.

κ is inaccessible and the infinitary language L_{κ,κ} satisfies the weak compactness theorem.

κ is inaccessible and the infinitary language L_{κ,ω} satisfies the weak compactness theorem.
A language L_{κ,κ} is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.
See also
References

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


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