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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.