Related papers: Are monochromatic Pythagorean triples unavoidable …

Recently S. Goswami proved that whenever the set $\mathbb N$ of natural numbers is finitely colored, the set $\{a, b, ab, b(a+1)\}$ is monochromatic which also established a variant of the long-standing Hindman's conjecture, which asks for…

We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least ck^8 points of P, for some constant c, contains at…

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…

It is important in drawing techniques to find combinations of two straight lines and their angle bisectors whose slopes are all rational numbers. This problem is reduced to solving the Diophantine equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$…

Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in…

In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there…

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring…

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree $\Delta$ can be 3-colored in such a way that each monochromatic component has at most $f(\Delta)$ vertices.…

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…

In this paper we prove that if $S$ is any finite configuration of points in $\mathbb{Z}^2$, then any finite coloring of $\mathbb{E}^2$ must contain uncountably many monochromatic subsets homothetic to $S$. We extend a result of Brown,…

Finite graphs that have a common chromatic polynomial have the same number of regular $n$-colorings. A natural question is whether there exists a natural bijection between regular $n$-colorings. We address this question using a functorial…

Even though four theorems are actually proved in this paper, two are the main ones,Teorems 1 and 3. In Theorem 1 we show that if a and be are odd squarefree positive integers satisfying certain quadratic residue conditions; then there…

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group…

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…

We prove that in every $2$-edge-colouring of $K_n$ there is a collection of $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, thus confirming a conjecture of Erd\H{o}s. We also prove a corresponding stability result, showing that…

In an article [3] published recently in this journal, it was shown that when k >= 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard. We consider the problem when k = 2. In regards to…

A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of…

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach,…

Arrangements of pseudolines are a widely studied generalization of line arrangements. They are defined as a finite family of infinite curves in the Euclidean plane, any two of which intersect at exactly one point. One can state various…

We study the infinite graph of $n$-dimensional rectangular grid that doesn't appear distance regular and the distance regular colorings of this graph, which are defined as the distance colorings with respect to completely regular codes. It…