Weakly compact cardinal

In mathematics, a cardinal κ is weakly compact iff for every function f: &kappa ² → {0, 1} there is a set of cardinality κ that is homogeneous for f.

Theorem

The following are equivalent for any uncountable cardinal κ:

1. κ is weakly compact
2. for every λ<κ, integer n, and function f: &kappaⁿ → λ there is a set of cardinality κ that is homogeneous for f
   
3. κ is inaccessible and every tree of height κ either has a path or a level of cardinality at least κ
   
4. Every linear order of cardinality κ has an ascending or a descending sequence of order type κ