Weakly compact cardinal

A weakly compact cardinal (WCC for short) is a certain type of large cardinals with many equivalent definitions, such as this one:
:Let \([x]^2\) be all the 2-element subsets of \(x\). Then an uncountable cardinal \(\alpha\) is weakly compact if and only if, for every function \(f: [\alpha]^2 \rightarrow \{0, 1\}\), there is a set \(S \subseteq \alpha\) such that \(|S| = \alpha\) and \(f\) maps every member of \([S]^2\) to either all 0 or all 1.
:More intuitively, any two-coloring of the edges of the complete graph \(K_\alpha\) contains a monochromatic subgraph isomorphic to \(K_\alpha\).
A WCC is always inaccessible and Mahlo. Thus they cannot be proven to exist in ZFC (assuming it is consistent), and ZFC + "there exists a WCC" is believed to be consistent.
The least WCC (if it exists) is sometimes called "the" weakly compact cardinal \(K\).
==Collapse==
To googologists, \(K\) and other WCCs are mostly useful through ordinal collapsing functions such as Rathjen's Ψ function, and ordinal notations associated to them. In ordinal collapsing functions, weakly inaccessible cardinals are used to diagonalise cardinals, weakly Mahlo cardinals are used to diagonalise weakly inaccessible cardinals, and the weakly compact cardinal is used to diagonalise weakly Mahlo cardinals. Therefore many googologists who do not know the definition of weakly compact cardinals often presume that there were a reasonable sequence \((\Omega,I,M,K,\ldots)\) and its diagonalisation which work in ordinal collapsing functions in a desired way, although there is not such a known sequence.
== See also ==
ja:弱コンパクト基数
Category:Transfinite ordinals