| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

3. Partition Regularity

A polynomial $P(x_1,\ldots,x_m)$ is partition regular if for any finite coloring $\mathbb{N}=C_1\cup\ldots\cup C_k$ there is some $n\in\{0,\ldots,k\}$ and some $a_1,\ldots,a_m\in\mathbb{N}$ such that $P(a_1,\ldots,a_m)=0$ and $a_i\in C_n$ for all $i=1,\ldots,m$.

Depending on the equation, one may ask for the sequence $(a_1,\ldots,a_m)$ above to be non-constant, or even injective. For linear polynomials, partition regularity is characterized by the following result.
Theorem.
(Rado 1933 [MR1545354]) A linear polynomial $c_1x_1+\ldots+c_mx_m$ is partition regular if and only if there is some $\emptyset\neq I\subseteq\{1,\dots,m\}$ such that $\sum_{i\in I}c_i=0$.


    1. Add remark

      Problem 3.1.

      [Martino Lupini] Is $x^2+y^2-z^2$ partition regular?
          A positive result is known for $2$-colorings, with a computer-assisted proof [MR3534782].

        •     A norm form is a polynomial constructed as follows. Fix $\mathbb{Q}$-linearly independent algebraic numbers $\omega_1,\ldots,\omega_N$, and let $K$ be the field they generate over $\mathbb{Q}$. Let $\sigma_1,\ldots,\sigma_d$ be the distinct embeddings of $K$ into the algebraic closure of $\mathbb{Q}$. Define \[ F(x_1,\ldots,x_n)=\prod_{i=1}^d\sum_{j=1}^N\sigma_i(\omega_j)x_j. \] Then $F(\bar{x})$ has rational coefficients and is not identically zero. In fact, \[ F(\bar{x})=\text{Norm}_{K/\mathbb{Q}}(\omega_1x_1+\ldots+\omega_Nx_N). \] A norm form equation is an equation of the form \[ F(\bar{x})=\pm\beta, \] where $\beta\in\mathbb{Z}^N$. A result of Schmidt [MR0308062] says that a norm form equation has finitely many solutions if and only if the corresponding norm form $F(\bar{x})$ is non-degenerate.

          Add remark

          Problem 3.2.

          [Shabnam Akhtari] Study partition regularity for degenerate norm form equations.

            •     A system of polynomials $\{f_1(x_1,\ldots,x_m),\ldots,f_t(x_1,\ldots,x_m)\}$ is a Ramsey family if for any finite coloring $\mathbb{N}=C_1\cup\ldots\cup C_k$, there are $a_1,\ldots,a_m\in\mathbb{N}$ and some $n\in\{1,\ldots,k\}$ such that $f_i(\bar{a})\in C_n$ for all $i=1,\ldots,t$.

              A classical result of Schur states that $\{x,y,x+y\}$ is a Ramsey family, and a longstanding question asked if $\{x+y,xy\}$ is a Ramsey family. This is answered by the following recent theorem.
              Theorem.
              (Moreira 2017 [MR3664819]) $\{x,x+y,xy\}$ is a Ramsey family.

              Add remark

              Problem 3.3.

              [?**?] Is $\{x,y,x+y,xy\}$ a Ramsey family?

                  Cite this as: AimPL: Nonstandard methods in combinatorial number theory, available at http://aimpl.org/nscombinatorial.