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For $k=2$, from this post A "geometrical" representation for Ramsey's theorem, how one can deduce the theorem from Constant $f:[\mathbb{N}]^2\to \{1,2\}$ (part 2), or by knowing that there exist infinite sequence of naturals $n_1,n_2,n_3,...$, and a sequence of $t_1,t_2,t_3,...$ where $t_n\in\{1,2\}$ for all $n$, and if $i<j$, then $f(\{n_i,n_j\})=t_i$.

How to represent set $H$ (set $H$ is from the first link)?

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up vote 2 down vote accepted

You’re practically done at this point.

Let $N_1=\{n_k:t_k=1\}$ and $N_2=\{n_k:t_k=2\}$. At least one of these sets must be infinite, so without loss of generality assume that $N_1$ is infinite; then $f(\{n_i,n_k\})=1$ for all $\{n_i,n_k\}\in[N_1]^2$.

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