A cute proof of Rainbow Ramsey Theorem

Just saw this cute little proof of the Rainbow Ramsey Theorem (an Anti-Ramsey-type Theorem). But before that, a colouring $\chi:[\omega]^{k}\to\omega$ is called $n$-bounded if $|\chi^{-1}(c)|\leq n$ for all colours $c\in\omega$. We say a set $H\in[\omega]^{\omega}$ is rainbow if $\chi|_{[H]^k}$ is an injection.

Rainbow Ramsey Theorem ($\textsf{RRT}^k_n$). Let $k,n\in\omega$ with $n\geq 1$, for any $n$-bounded colouring $\chi:[\omega]^{k}\to\omega$, there exists a set $H$ which is rainbow under $\chi$.

Proof. It’s a simple special case of the much general Canonical RT ($\infty$-version), but here is a sweet little application of Infinitary RT. Let $\prec$ be a well-ordering on $[\omega]^k$. Consider the colouring $COL:[\omega]^{k}\to n$ defined by $COL(x):=|\{y\in[\omega]^k:, y\prec x\text{ and } \chi(y)=\chi(x)\}|.$
By $\textsf{IRT}$, $\exists H\in[\omega]^{\omega}$ homogeneous under $COL$. But notice that for distinct $x, y\in[H]^{k}$, $COL(x)=COL(y)$ and either $x\prec y$ or $y\prec x$ forces $\chi(x)\neq \chi(y)$ (FTSOC assume $\chi(x)=\chi(y)$, then $x\prec y\implies COL(y)>COL(x)$ and $y\prec x\implies COL(x)>COL(y)$, $\Rightarrow\Leftarrow$). From this it follows that $H$ is rainbow under $\chi$.