A weakly compact cardinal (WCC) is a certain type of large cardinal 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 \(f: [\alpha]^2 \mapsto \{0, 1\}\), 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 \(K_\alpha\) as a subgraph.

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 \(K\). To googologists, \(K\) and other WCCs are mostly useful through ordinal collapsing functions.

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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