Tree property ($\mathrm{TP}(\kappa )$)

For an infinite cardinal $\kappa$, the tree property says that every tree of height $\kappa$ with nonempty levels of cardinality less than $\kappa$ has a cofinal branch. In the all-trees-of-size-$\kappa$ convention it also forces regularity: disjoint chains of lengths cofinal in a singular $\kappa$ form a counterexample. It does not, by itself, imply strong inaccessibility. A strong limit cardinal with the tree property satisfies the self-partition relation by applying it to the end-homogeneous coloring tree. At a strongly inaccessible cardinal this is the usual weak-compactness characterization.