Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 130 / 1 / c

Let $c:\mathbb{N}\to [k]$ be the given finite coloring. Color each $t$-element subset $\{y_1,\dots ,y_t\}$ of $\mathbb{N}$ byBy Ramsey's theorem there is an infinite set $Y$ whose $t$-element subsets all receive the same induced color. Enumerate it increasingly as $x_1<x_2<\cdots$. Then every sum $x_{i_1}^2+\cdots +x_{i_t}^2$ with $i_1<\cdots <i_t$ has that color. The argument works for every positive integer $t$; primality is not needed for this part.

Solved by gpt-5.6-sol high.

No. It is enough to take the prime number $t=2$. By the monochromatic sums-and-products obstruction, there is a finite coloring $\chi$ of $\mathbb{N}$ for which no infinite set has all its pairwise sums and pairwise products in one color. Refine $\chi$ by also recording the parity of the 2-adic valuation.

If a sequence $(x_i)$ made both requested families monochromatic, put $y_i=x_i^2$. If the set of distinct $y_i$ were infinite, an injective subsequence would make all pairwise sums $y_i+y_j$ and products $y_i{y}_j$ monochromatic under $\chi$, a contradiction. Otherwise some $y=x_i^2$ occurs infinitely often. Two occurrences give the sum $2y$ and the product $y^2$, butbecause $v_2(y)$ is even. The refining colors differ, another contradiction.

Solved by gpt-5.6-sol high.