= Solution
Write $\mathbb N=\{1,2,\ldots\}$ and let $\chi$ be the given <finite coloring> of the <edges> of the <complete graph> on the <positive integers>. Construct an increasing <sequence> $(v_i)$ and nested <infinite sets> $A_0\supset A_1\supset\cdots$ as follows. Set $A_0=\mathbb N$. Having chosen $A_{i-1}$, take $v_i=\min A_{i-1}$, partition $A_{i-1}\setminus\{v_i\}$ by the color of $\{v_i,w\}$, and use the <infinite pigeonhole principle> to choose an <infinite set> $A_i$ on which this color is constant, say $r_i$. Every element of $A_i$ exceeds $v_i$.
The <infinite pigeonhole principle> applied again to $(r_i)$ gives an <infinite set> of indices $I$ and a color $r$ such that $r_i=r$ for all $i\in I$. For $i<j$ in $I$, nesting gives $v_j\in A_i$, so $\chi(\{v_i,v_j\})=r$. Thus the <infinite set> $X=\{v_i:i\in I\}$ is a <monochromatic> <complete graph>:
$$
\boxed{\chi\big|_{X^{(2)}}\equiv r.}
$$
This proves the required infinite case of <Ramsey's theorem>; finiteness of the color set is used at both applications of the <infinite pigeonhole principle>.
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