Weakly compact cardinals

In ZFC a cardinal kappa is weakly compact when it is inaccessible in the sense of Inaccessible and Mahlo cardinals and has the tree property of κ-trees and the tree property: every tree of height kappa with fewer than kappa nodes at each level has a cofinal branch. Cardinal comparisons use The Axiom of Choice.

The partition notation $\kappa ⟶(\kappa {)}_{\mu }^2$ has the meaning in Partition arrows and homogeneous sets: each coloring of unordered pairs by the nonzero cardinal mu has a homogeneous subset of cardinality kappa. At an inaccessible, the tree property is equivalent to the two-color instance and, equivalently, to the simultaneous assertion of these arrows for every nonzero $\mu <\kappa$; that result is proved subsequently and is not assumed here. No equivalence is claimed for any single arbitrary $\mu$, nor for colors $\mu \ge \kappa$. Inaccessibility is part of this definition, so the tree property alone at a successor cardinal does not establish weak compactness. This is a property of a cardinal, with no asserted existence and no witness-selection definition to justify.