= Solution
Pull the given <finite coloring> $\chi:\mathbb N\to[k]$ back to a <finite coloring> of unordered pairs by putting
$$
\widetilde\chi(\{a,b\})=\chi(ab^2)\qquad(a<b).
$$
The ordering $a<b$ makes this a well-defined <function> on the <edges> of a <complete graph>. By <Ramsey's theorem>, some <infinite set> $X=\{x_1<x_2<\cdots\}$ has all its pairs of one color $r$ under $\widetilde\chi$. Consequently
$$
\boxed{\chi(x_i x_j^2)=r\qquad(i<j).}
$$
Possible coincidences between these products cause no difficulty, since the original <finite coloring> assigns each <positive integer> a single color.
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