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An arithmetic function is a function whose domain is the positive integers, usually taking values in the complex numbers.

For positive integers $k,s,\ell$, some $N$ ensures that every $k$-color finite coloring of $[N]$ contains a monochromatic setIt follows from the Van der Waerden theorem by mathematical induction on the number of colors. If $M$ is a bound for $k-1$ colors, take a long one-color arithmetic progression of length $(\ell -1)M+1$ and step $d$. Either one of $sd,2sd,\dots ,Msd$ has its color, giving the result immediately, or these multiples use only $k-1$ colors. Pull back their finite coloring to $[M]$ and use mathematical induction, then multiply the resulting configuration by $sd$. Enlarge the ambient finite integer interval to include all these multiples. The cases $k=1$ and $\ell =1$ are immediate. The parameter $s$ makes this useful for positive monochromatic solutions of a partition regular equation with arbitrary integer coefficients.

If a rational matrix is a partition regular matrix, then for each fixed number of colors $k$ some finite integer interval $[T]$ already forces a monochromatic positive solution. Otherwise the solution-free finite colorings of successive intervals form a finitely branching tree with every level nonempty. The König infinity lemma gives an infinite branch, contradicting partition regularity. More generally, this argument applies to any family of configurations each using finitely many positive integers.

The constant-one arithmetic function has $1(n)=1$ for every positive integer $n$. Its Dirichlet convolution with $f$ is the divisor sum $(1\ast f)(n)={\sum }_{d\mid n}f(d)$.

Extend a finite coloring of the positive integers to a point $x$ of a two-sided full shift. A minimal subsystem of its forward orbit closure, together with the proximal-minimal existence theorem, supplies a minimal point $y$ proximal to $x$. Put $q=y(0)$. If all sums in $\{0\}\cup \mathrm{FS}(a_1,\dots ,a_r)$, the augmented finite-sums set, have color $q$ in $y$, their coordinate constraints define a cylinder set containing $y$. The joint return lemma for a proximal minimal pair chooses a new positive term so that all new sums have color $q$ in both $x$ and $y$. Mathematical induction gives an infinite monochromatic finite-sums set. The new term can exceed the sum of all previous terms, giving unique representations. The proximal-minimal existence theorem is a substantive input to this proof.

For every dimension $m$, every finite coloring of the positive integers contains $x_1,\dots ,x_m$ for which all nonempty sums of distinct $x_i$ have one color. This is the case $p=c=1$ of the monochromatic m-p-c set theorem.

If $A$ is a finite $K$-approximate group, thenfor every positive integer $m$, where $A^2\subseteq XA$ and $|X|\le K$. This follows immediately by mathematical induction.

A Hilbert $n$-cube in the positive integers is a setwhere $a,x_1,\dots ,x_n$ are positive integers.

Every finite coloring of the positive integers contains a monochromatic Hilbert cube of every prescribed finite dimension.

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.

An ideal $I$ of a Noetherian local ring $(A,\mathfrak{m})$ is an ideal of definition when $A/I$ has Krull dimension zero. Equivalently, $I$ is $\mathfrak{m}$-primary, so $\sqrt{I}=\mathfrak{m}$ and ${\mathfrak{m}}^r\subseteq I$ for some positive integer $r$.

A positive integer in this set has no prime factor equal to two or congruent to three modulo four. Its prime factors all lie in the residue class one modulo four. For example $5\cdot 13=65$ belongs, while $3^2=9$ does not, even though $9$ itself is congruent to one modulo four. A half-dimensional interval sieve bounds the count in an interval of length $z$ by $O(z/\sqrt{\log z})$.

For an odd prime $p$, the least quadratic nonresidue $n(p)$ is the smallest positive integer whose nonzero residue class is a quadratic nonresidue modulo $p$.

For every positive $m,p,c$, every finite coloring of the positive integers contains a monochromatic M-p-c set.

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 linear map $T:V\to V$ is nilpotent if $T^k=0$ for some positive integer $k$.

Using the axiom of choice, enumerate all infinite subsets $A_{\alpha }$ of the positive integers in order type $2^{{\aleph }_0}$. By transfinite recursion, choose two previously unused infinite subsets $X_{\alpha },Y_{\alpha }$ of each $A_{\alpha }$. This is possible because each $[A_{\alpha }{]}^{\omega }$ has cardinality $2^{{\aleph }_0}$ and fewer choices have been made at stage $\alpha$. The set $\{X_{\alpha }\}$ is not a Ramsey set of infinite subsets, since every $[A_{\alpha }{]}^{\omega }$ meets both it and its complement.

A matrix $A$ over the rational numbers is partition regular when every finite coloring of the positive integers admits a monochromatic vector $x$ satisfying $Ax=0$, with every coordinate a positive integer. Repeated coordinates are permitted.