Related papers: On minimal colorings without monochromatic solutio…

We study the number of monochromatic solution to linear equation in $\{1,\dots,n\}$ when we color the set by at least three colors. We consider the $r$-commonness for $r\geq 3$ of linear equation with odd number of terms, and we also prove…

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge…

We show that for every coloring of the rationals into finitely many colors, one of the colors contains a set of the form $\{x,y,xy,x+y\}$ for some nonzero $x$ and $y$.

There exists a minimum integer $N$ such that any 2-coloring of $\{1,2,...,N\}$ admits a monochromatic solution to $x+y+kz =\ell w$ for $k,\ell \in \mathbb{Z}^+$, where $N$ depends on $k$ and $\ell$. We determine $N$ when $\ell-k \in…

Given an equation, the integers $[n] = \{1, 2, \dots, n\}$ as inputs, and the colors red and blue, how can we color $[n]$ in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is…

Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero constant term. We show that for any finite coloring of $\mathbb{Q}$, there are non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a set of…

Let $f_r(k)$ be the smallest positive integer $n$ such that every $r$-coloring of $\{1,2,...,n\}$ has a monochromatic solution to the nonlinear equation \[1/x_1+\cdots+1/x_k=1/y,\] where $x_1,...,x_k$ are not necessarily distinct. Brown and…

We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of…

Let $G$ be a finite abelian group with exponent $n$, and let $r$ be a positive integer. Let $A$ be a $k\times m$ matrix with integer entries. We show that if $A$ satisfies some natural conditions and $|G|$ is large enough then, for each…

We show that for any two linear homogenous equations $\mathcal{E}_0,\mathcal{E}_1$, each with at least three variables and coefficients not all the same sign, any 2-coloring of $\mathbb{Z}^+$ admits monochromatic solutions of color 0 to…

It was shown that any $\mathbb{Z}$-colorable link has a diagram which admits a non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, we consider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings on…

The following question was asked by Prendiville: given an $r$-colouring of the interval $\{2, \dotsc, N\}$, what is the minimum number of monochromatic solutions of the equation $xy = z$? For $r=2$, we show that there are always…

The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let $\mathcal{E}$ be a linear equation. Denote by $\operatorname{R}_r(\mathcal{E})$ the minimal integer, if it exists, such that any $r$-coloring of…

Let $n\in \mathbb{N}$, $R$ be a binary relation on $[n]$, and $C_1(i,j),\ldots,C_n(i,j) \in \mathbb{Z}$, for $i,j \in [n]$. We define the exponential system of equations $\mathcal{E}(R,(C_k(i,j)_{i,j,k})$ to be the system \[…

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a…

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more…

Consider the set $\{1,2,\dots,n\} = [n]$ and an equation $eq$. The rainbow number of $[n]$ for $eq$, denoted $\operatorname{rb}([n],eq)$, is the smallest number of colors such that for every exact $\operatorname{rb}([n], eq)$-coloring of…

We determine the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included are complete classifications of such $\mathbb{Z}$-colorings and of such…

We show that the minimal number of colors for all effective $n$-colorings of a link with non-zero determinant is at least $1+\log_2 n$.

For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is…