Related papers: On the minimum number of monochromatic solutions t…

Green and Lindqvist proved that for any 2-colouring of $\mathbb{N}$, there are in\-fi\-ni\-tely many monochromatic solutions to $x+y=z^2$. In fact, they showed the existence of a monochromatic solution in every interval $[N,cN^8]$ with…

For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t_{k-1} \geq 3$, the generalized Schur…

Let k, r, s in the natural numbers where r \geq s \geq 2. Define f(s,r,k) to be the smallest positive integer n such that for every coloring of the integers in [1,n] there exist subsets S_1 and S_2 such that: (a) S_1 and S_2 are…

For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence…

Given a linear equation $\mathcal{E}$, the $k$-color Rado number $R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of $\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The degree of regularity…

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…

For positive integers $a_1,a_2,...,a_m$, we determine the least positive integer $R(a_1,...,a_m)$ such that for every 2-coloring of the set $[1,n]={1,...,n}$ with $n\ge R(a_1,...,a_m)$ there exists a monochromatic solution to the equation…

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $S_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $\chi:[1,n] \rightarrow \{0,1,\dots,r-1\}$ admits a solution to $\sum_{i=1}^{k-1} x_i = x_k$…

We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$ minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers…

Let $r$, $m$ and $k\geq 2$ be positive integers such that $r\mid k$ and let $v \in \left[ 0,\lfloor \frac{k-1}{2r} \rfloor \right]$ be any integer. For any integer $\ell \in [1, k]$ and $\epsilon \in \{0,1\}$, we let…

Liu, Pach and S\'andor recently characterized all polynomials $p(z)$ such that the equation $x+y=p(z)$ is $2$-Ramsey, that is, any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. In this paper,…

Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component,…

The multi-fold chromatic number of the plane $\chi_m$ is the smallest number of colors $k$, sufficient to color each point of the Euclidean plane in exactly $m$ colors, so that for any pair of points at a unit distance from each other, two…

Let $D$ be a set of positive integers. A $D$-diffsequence of length $k$ is a sequence of positive integers $a_1 < \cdots < a_k$ such that $a_{i+1}-a_i\in D$ for $i=1,\ldots,k-1$. For $D=\{2^i\mid i\in \mathbb{Z}_{\ge 0}\}$, it is known that…

In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this…

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…

Attending to an open problem in the literature stated by Mirzakhani and Vondr\'ak, we give a lower bound of the number of non-monochromatic simplices for Sperner labelings of the vertices of a triangulation of a given $ k$-simplex with…

For an integer $r \ge 2$ and an order $n \equiv 1, 3 \pmod{6}$, write $\delta_r(n)$ for the minimum, over all $r$-colourings $\chi : \binom{[n]}{3} \to [r]$, of $\max_{\mathcal{S}} \mathrm{disc}(\mathcal{S}, \chi)$, where the maximum is…

Gy\'arf\'as and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of $K_{n,n}$ there exists a monochromatic path on at least $2\lceil n/2\rceil$ vertices, and this is tight. We prove a stability version…

The Ramsey number $R(s,t)$ is the least integer $n$ such that any coloring of the edges of $K_n$ with two colors produces either a monochromatic $K_s$ in one color or a monochromatic $K_t$ in the other. If $s=t$, we say that the Ramsey…