= Simultaneous coefficient patterns from a homogeneous four-set colouring
{title2=$x_i=u_{2i-1}+u_{2i},\quad y_i=(u_1+u_2)/2+2u_{i+2}$}
Take increasing even $u_i$ for which every ordered four-index expression $u_i+u_j+2u_k+2u_l$ has one colour. The <Ramsey theorem for r-sets> supplies them. Set $x_i=u_{2i-1}+u_{2i}$ and $y_i=(u_1+u_2)/2+2u_{i+2}$. Then both $x_i+2x_j$ and $y_i+y_j$ use the same ordered coefficient pattern $(1,1,2,2)$, so their union is monochromatic. The shared half-prefix makes the unweighted pair family possible without asking the two families to use the same generating sequence.
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