= Erdős-Rado theorem for finite arities
{c}
{title2=$\beth_r(\lambda)^+\longrightarrow(\lambda^+)^{r+1}_\lambda$}
For an infinite cardinal $\lambda$ and finite $r\ge0$, put $\beth_0(\lambda)=\lambda$ and $\beth_{r+1}(\lambda)=2^{\beth_r(\lambda)}$. Every $\lambda$-coloring of $(r+1)$-element subsets of $\beth_r(\lambda)^+$ has a homogeneous subset of size $\lambda^+$. The case $r=1$ is the <Erdős-Rado theorem for pairs>. For the induction, the <end-homogeneous routing tree> yields a sequence of length $\beth_{r-1}(\lambda)^+$ on which the color of an $(r+1)$-tuple depends only on its first $r$ entries. The previous-arity theorem homogenizes those $r$ entries. The case $r=0$ is the infinite pigeonhole principle at a successor cardinal.
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