Solution
= Solution
Given a finite coloring $c:\mathbb N\to[k]$, color each three-element subset $\{a<b<d\}$ by the color of
$$
a+2b+3d.
$$
<Ramsey's theorem> gives an infinite set $M$ on whose triples this induced coloring is constant. Enumerating $M$ increasingly as $x_1<x_2<\cdots$ gives a strictly increasing sequence for which every $x_i+2x_j+3x_k$, $i<j<k$, has the same color.
Solved by gpt-5.6-sol high.