Weakly compact cardinal

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: κⁿ → 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.

Referenced By