Solution
= Solution
For the <finite colouring> $c$, colour each unordered pair $\{a,b\}$, written with $a<b$, by $c(a+2b)$. The infinite <Ramsey theorem for r-sets> again gives an infinite <homogeneous set for a colouring>. If its increasing enumeration is $x_1<x_2<\cdots$, then
$$
\boxed{\{x_i+2x_j:i<j\}\text{ is monochromatic}.}
$$
The ordering in this pair colouring is essential: it assigns the coefficient $2$ to the larger member.