Related papers: Schur Number Five

Schur's theorem states that in any $k$-colouring of the set of integers $[n]$ there is a monochromatic solution to $a+b=c$, provided $n$ is sufficiently large. Abbott and Wang studied the size of the largest subset of $[n]$ such that there…

Suppose that each number $1,2,...,N$ has one of n colours assigned. We show that if there are no monochromatic solutions to the equation $x_1+x_2+x_3=y_1+y_2$, then $N=O((n!)^{1/2})$, improving upon a result of Cwalina and Schoen. Further,…

A packing $k$-coloring is a natural variation on the standard notion of graph $k$-coloring, where vertices are assigned numbers from $\{1, \ldots, k\}$, and any two vertices assigned a common color $c \in \{1, \ldots, k\}$ need to be at a…

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate…

A set $A$ of integers is said to be Schur if any two-colouring of $A$ results in monochromatic $x,y$ and $z$ with $x+y=z$. We study the following problem: how many random integers from $[n]$ need to be added to some $A\subseteq [n]$ to…

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,…

A classical result in combinatorial number theory states that the largest subset of $[n]$ avoiding a solution to the equation $x+y=z$ is of size $\lceil n/2 \rceil$. For all integers $k>m$, we prove multicolored extensions of this result…

Graham, R\"odl, and Ruci\'nski originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first $n$ integers. This question was subsequently resolved independently…

Schur's Theorem states that, for any $r \in \mathbb{Z}^+$, there exists a minimum integer $S(r)$ such that every $r$-coloring of $\{1,2,\dots,S(r)\}$ admits a monochromatic solution to $x+y=z$. Recently, Budden determined the related…

This paper is concerned with the problem of finding $n$ distinct squares such that, on excluding any one of them, the sum of the remaining $n-1$ squares is a square. While parametric solutions are known when $n=3$ and $n=4$, when $n > 4$,…

We present a new proof of the known fact that the chromatic number of the plane is at least 5. The main difference of this proof is that it can be verified manually without the help of the computer.

Given a natural $n$, we construct a two-coloring of $\mathbb{R}^n$ with the maximum metric satisfying the following. For any finite set of reals $S$ with diameter greater than $5^{n}$ such that the distance between any two consecutive…

The solution to the problem of finding the minimum number of monochromatic triples $(x,y,x+ay)$ with $a\geq 2$ being a fixed positive integer over any 2-coloring of $[1,n]$ was conjectured by Butler, Costello, and Graham (2010) and…

A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, the minimum $N$ such that any $r$-coloring of a…

The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\softO$) ignores logarithmic factors)…

In 1977, Erd\H{o}s asked the following question: for any integers $t,n \in \mathbb{N}$, if $G_1 , \dots , G_n$ are complete graphs such that each $G_i$ has at most $n$ vertices and every pair of them shares at most $t$ vertices, what is the…

Let $\mathcal{C}_k(n)$ be the family of all connected $k$-chromatic graphs of order $n$. Given a natural number $x\geq k$, we consider the problem of finding the maximum number of $x$-colorings among graphs in $\mathcal{C}_k(n)$. When…

Suppose that $n \ge 2$, and we wish to plant $k$ different types of trees in the squares of an $n \times n$ square grid. We can have as many of each type as we want. The only rule is that every pair of types must occur in an adjacent pair…

For G a finite non-Abelian group we write c(G) for the probability that two randomly chosen elements commute and k(G) for the largest integer such that any k(G)-colouring of G is guaranteed to contain a monochromatic quadruple (x,y,xy,yx)…

A square coloring of a graph $G$ is a coloring of the square $G^2$ of $G$, that is, a coloring of the vertices of $G$ such that any two vertices that are at distance at most $2$ in $G$ receive different colors. We investigate the complexity…