= Ordinal partition-bound function
{title2=$f(\alpha)=\min\{\beta\geq\alpha:\forall\nu<\alpha\ (\beta\to(\alpha)^2_\nu)\}$}
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 $f(\alpha)\geq\alpha$. 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.
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