Partition arrows and homogeneous sets

Let $\kappa ,\lambda$ be cardinals as in Cardinal (initial ordinal) and cardinality, let $n<\omega$, and let $r$ be a nonzero cardinal of colors. Write $[X{]}^n=\{u\subseteq X:\mathrm{\mid }u\mathrm{\mid }=n\}$. A set $H\subseteq X$ is homogeneous for $c:[X{]}^n\to r$ if some $i<r$ satisfies $c(u)=i$ for all $u\in [H{]}^n$. The cardinal partition arrow

$\kappa ⟶(\lambda {)}_r^n$

means that every map $c:[\kappa {]}^n\to r$ admits such an $H$ with $\mathrm{\mid }H\mathrm{\mid }=\lambda$. The negated arrow asserts that some coloring has no such homogeneous set. Here the size target is a cardinal; an ordinal order-type target would require a separately stated convention.

The parameter $n$ is fixed before the coloring is quantified; this is not a simultaneous homogeneity assertion for all finite arities. The color cardinal $r$ may be infinite, while the infinite Ramsey theorem below restricts it to a positive finite integer. If $r=1$, every subset is homogeneous. If $n=0$, $[H{]}^0=\{\mathrm{\varnothing }\}$ for every $H$, so every subset is homogeneous, with color $c(\mathrm{\varnothing })$. If $\mathrm{\mid }H\mathrm{\mid }<n$ and $n>0$, the homogeneous requirement is vacuous, with any color $i<r$. These conventions include $H=\mathrm{\varnothing }$. We exclude $r=0$ to avoid vacuous nonexistence of colorings. If $\lambda >\kappa$, the arrow fails: the constant-zero coloring exists since $r\mathrm{\ne }0$, but there is no subset of cardinality $\lambda$.