About: Weakly compact cardinal   Sponge Permalink

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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.

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  • Weakly compact cardinal
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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.
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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.
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