 Solution

Let $c:\mathbb{N}\to \{1,\dots ,r\}$ be the given finite colouring, with $\mathbb{N}=\{1,2,\dots \}$. 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, soThis application of the infinite Ramsey theorem for r-sets uses only unordered pairs.