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For every finite $S\subseteq {\mathbb{N}}^d$, every finite coloring of ${\mathbb{N}}^d$ contains a monochromatic homothetic copy $a+rS$. The theorem follows from the Hales-Jewett theorem by coloring a word according to the sum of the points indexed by its letters.

Every finite coloring of the positive integers has an infinite sequence $(x_i)$ for which all nonempty finite sums of distinct terms have one color.

There is a finite coloring of the positive integers for which no infinite set has all pairwise sums and pairwise products in one color. Refining this coloring by the parity of the 2-adic valuation also prevents a constant infinite sequence from evading the obstruction.

A rational matrix $A$ is partition regular when every finite coloring of the positive integers admits a nonzero monochromatic vector $x$ satisfying $Ax=0$.

Ramsey's theorem for $r$-sets says that every finite coloring of $[\mathbb{N}{]}^{(r)}$ has an infinite monochromatic set. We prove it by mathematical induction on $r$. The case $r=1$ is the infinite pigeonhole principle. Suppose the result holds for $r-1$, and let $c:[\mathbb{N}{]}^{(r)}\to [k]$. Choose $a_1$, then use the induction hypothesis on the coloring $F\mapsto c(F\cup \{a_1\})$ to obtain an infinite set $M_1$ on which this color is constant, say $d_1$. Inductively chooseso that $c(F\cup \{a_j\})=d_j$ for every $F\in [M_j{]}^{(r-1)}$. Some color $d$ occurs for infinitely many $d_j$. If $j_1<j_2<\cdots$ are the corresponding indices, every $r$-set from $\{a_{j_1},a_{j_2},\dots \}$ has color $d$: take its least-indexed element $a_{j_s}$, after which its other $r-1$ elements lie in $M_{j_s}$. This proves the theorem.

Solved by gpt-5.6-sol high.

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.

Ramsey theory studies the ordered configurations that every finite coloring of a sufficiently large structure must contain.

Every finite coloring of the positive integers contains monochromatic arithmetic progressions of every prescribed finite length.