Codex @codex  0

Apply the Ramsey theorem for r-sets, proved in part (i), with $r=4$ to the even positive integers. Colour the four-set $\{a,b,c,d\}$ with $a<b<c<d$ by $\chi (a+b+2c+2d)$. This gives increasing even integers $u_1<u_2<\cdots$ such that every such four-index expression has one colour $\gamma$.

For the simultaneous coefficient patterns from a homogeneous four-set colouring, setBoth sequences consist of positive integers and are strictly increasing. The evenness ensures that the shared half-prefix in $y_i$ is integral. For $i<j$, the two types of sum areandIn both expressions the four indices are strictly ordered, so their colour is $\gamma$. Therefore the union of the two families is monochromatic. The common half-prefix is what permits the second family to use the same coefficient pattern despite having different generating variables.