Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 130 / 3 / c

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.

No. The incompatibility part of the Milliken–Taylor theorem provides a finite coloring of $\mathbb{N}$ for which the systems associated with the nonproportional compressed coefficient vectors $(1)$ and $(1,2,3)$ cannot be monochromatic in the same color. The first system is the finite-sums setwhile singleton separated blocks in the second include every $x_i+2x_j+3x_k$ with $i<j<k$. Thus a sequence satisfying the proposed union would contradict that finite coloring.

Solved by gpt-5.6-sol high.