Weakly compact cardinal
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 uncountable cardinals:
(a) κ is weakly compact
(b) for every λ<κ, integer n, and function f: &kappan → λ there is a set of cardinality κ that is homogeneous for f
(c) κ is inaccessible and every tree of height κ either has a path or a level of cardinality at least κ
(d) Every linear order of cardinality κ has an ascending or a descending sequence of order type κ
