= Solution
Let $c:\mathbb N\to\{1,\ldots,r\}$ be the given <finite colouring>, with $\mathbb N=\{1,2,\ldots\}$. Give each unordered pair $\{a,b\}$ the colour $c(a+b)$. The infinite <Ramsey theorem for r-sets>, in its two-element case, gives an infinite <homogeneous set for a colouring> $X$. Enumerate it as $x_1<x_2<\cdots$. Every distinct-index sum $x_i+x_j$ has the common pair colour, so
$$
\boxed{\{x_i+x_j:i\ne j\}\text{ is monochromatic}.}
$$
This application of the infinite <Ramsey theorem for r-sets> uses only unordered pairs.
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