 Solution

Given a finite coloring $c:\mathbb{N}\to [k]$, color each three-element subset $\{a<b<d\}$ by the color ofRamsey'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.