= End-homogeneous routing tree
{title2=$|T_\alpha|\le\lambda^{|[\alpha]^r|}$}
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.
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