 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 $x_1<x_2<\cdots$, thenThe ordering in this pair colouring is essential: it assigns the coefficient $2$ to the larger member.