Ordinal partition-bound function ($f(\alpha )=\min \{\beta \ge \alpha :\forall \nu <\alpha (\beta \to (\alpha {)}_{\nu }^2)\}$)

The colors $\nu$ here range over nonzero cardinals. The Erdős-Rado theorem for pairs makes this function total, and target and color monotonicity make it nondecreasing, with $f(\alpha )\ge \alpha$. If an uncountable cardinal $\kappa$ is closed below itself under $f$, then it is strong limit. Indeed, coloring distinct binary strings of length $\lambda <\kappa$ by their first differing coordinate uses $\lambda$ colors and has no monochromatic triangle, so $f(\lambda +1)>2^{\lambda }$. If $\kappa$ also has the tree property, the end-homogeneous tree for any $\nu <\kappa$ has levels of size below $\kappa$ and a cofinal branch; regularity makes one color occur $\kappa$ times. Thus $f(\kappa )=\kappa$. This distinguishes function closure plus the tree property from the tree property alone.