= Solution
Let $c:\mathbb N\to[k]$ be the given <finite coloring>. Color each $t$-element subset $\{y_1,\ldots,y_t\}$ of $\mathbb N$ by
$$
c(y_1^2+\cdots+y_t^2).
$$
By <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.
Back to article page
