End-homogeneous routing tree ($|T_{\alpha }|\le {\lambda }^{|[\alpha {]}^r|}$)

For a coloring of $(r+1)$-subsets of an ordinal, route a vertex through the least earlier vertices matching its colors on every $r$-subset of previously chosen ancestors. The path stops when the vertex itself is selected. Common path prefixes agree, yielding a set-theoretic tree; a node at level $\alpha$ is determined by its color table on $[\alpha {]}^r$. Along a branch, the color of an $(r+1)$-tuple is independent of its last entry. If the domain has size $(2^{\mu }{)}^{+}$ and $\lambda \le \mu$, levels below ${\mu }^{+}$ have size at most $2^{\mu }$; counting all those levels forces a branch of length ${\mu }^{+}$. If the domain is a regular strong-limit cardinal $\kappa$, those levels instead have size below $\kappa$, and the tree property supplies a cofinal branch when needed.