 Erdős-Rado theorem for finite arities

For an infinite cardinal $\lambda$ and finite $r\ge 0$, 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.