= Closed elementary-submodel construction of an end-homogeneous sequence
Put $\theta=(2^\mu)^+$ for an infinite <cardinal> $\mu$. Choose $M\prec H_\chi$ of size $2^\mu$, containing all <ordinals> below $\mu^+$, and closed under externally given <sequences> of length at most $\mu$. Such a model is built by $\mu^+$ successive <Skolem hull> closures, since $(2^\mu)^\mu=2^\mu$. For a coloring of $[\theta]^{r+1}$ with at most $\mu$ colors in $M$, put $\beta=\sup(M\cap\theta)<\theta$. Recursively choose $x_\alpha\in M\cap\theta$ for $\alpha<\mu^+$, above all earlier points, so that every earlier $r$-tuple has the same color with $x_\alpha$ as with $\beta$. There are at most $\mu$ constraints, their code belongs to $M$ by closure, and $\beta$ witnesses their consistency. The <elementary embedding> supplies the required witness in $M$. The resulting <sequence> is end-homogeneous, providing the induction step in the <Erdős-Rado theorem for finite arities>.
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