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.