Given a natural number n, let $ c:[X]^n \to \omega $ be a coloring by arbitrary many colors, where $X$ is an infinite countable set. Then there exists an infinite subset $ H $ of $ X $ for which the restriction of $ c $ to $ [H]^n $ is canonical. Meaning that for such restriction we have a fixed set $I\subseteq \lbrace 1,2,3,\ldots ,n \rbrace$ such that : $c(x_1\leq \cdots \leq x_n)=c(y_1\leq \cdots \leq y_n)$ if and only if $x_i=y_i$ for $i\in I$.
I'm looking for a proof of this, but I've been unable to find it. I think it might be analogous to the proof for Theorem 4.8, p. 1367 in Grahams Handbook of Combinatorics, Volume 1. which reads:
Theorem 4.8. (Canonical Lemma, Erdős and Rado, 1952) For every choice of positive integers $p$, $n$ there exists $N(p,n)$ such that for every set $ X $, $ |X| \geq N(p,n)$ and for every coloring $c:[X]^p \to \mathbb{N}$ (i.e. coloring by arbitrary many colors) there exists a set $ Y \subseteq X $, $ |Y|\geq n $, such that the coloring $ c $ restricted to the set $ [Y]^p $ is canonical.
Here a coloring $c$ is said to be canonical if there exists a set $\omega\subseteq p$ such that $$c(x_1<\dots<x_p)=c(x_1'<\dots<x_p'\text{ iff } x_i=x_i'\text{ for } i\in\omega.$$ (Thus there are $2^p$ canonical coloring patterns.)
But a proof of this isn't provided in the text.