Related papers: Odd distances in colourings of the plane

We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial $f$ with integer coefficients and positive leading coefficient,…

We present an alternate proof of the fact that given any 4-coloring of the plane there exist two points unit distance apart which are identically colored.

We prove that if one colors each point of the Euclidean plane with one of five colors, then there exist two points of the same color that are either distance $1$ or distance $2$ apart.

We give a full, correct proof of the following result, earlier claimed by Erd\H{o}s and Komj\'ath. If the Continuum Hypothesis holds then there is a coloring of the plane with countably many colors, with no monocolored right triangle.

A measure theoretic approach of the problem that there exits a finite unit-distance graphs in the plane that are not five (or four) colorable.

In this note, we prove that any 2-coloring of the plane contains 4 points of the same color forming a rhombus with unit sides and non-unit diagonals, answering a question of Axenovich, Liu, and the second author.

The chromatic number of the plane problem asks for the minimum number of colors so that each point of the plane can be assigned a single color with the property that no two points unit-distance apart are identically colored. It is now known…

Two vertices of an odd-distance graph are connected by an edge if and only if their Euclidean distance is an odd integer. We construct a 6-chromatic odd-distance graph in the plane.

We prove that for any $\ell_p$-norm in the plane with $1<p<\infty$ and for every infinite $\mathcal{M} \subset \mathbb{R}^2$, there exists a two-colouring of the plane such that no isometric copy of $\mathcal{M}$ is monochromatic. On the…

We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously…

We generalize a result of Tibor Gallai as follows: for any finite set of points $\mathcal{S}$ in the plane, if the plane is colored in finitely many colors, then there exist $2^{\aleph_0}$ monochromatic subsets of the plane homothetic to…

We provide a human-verifiable proof that, in a certain sense, the chromatic number of the plane is exactly 7.

We present a proof, using spectral techniques, that there is no finite measurable coloring of the odd-distance graph.

A conjecture of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus states that, with the exception of equilateral triangles, any two-coloring of the plane will have a monochromatic congruent copy of every three-point…

The paper proves the existence of monochrome standard simplexes of a given volume on a multidimensional rational lattice painted in a finite number of colors.

In the paper we prove, in particular, that for any measurable coloring of the euclidian plane into two colours there is a monochromatic triangle with some restrictions on the sides. Also we consider similar problems in finite fields…

We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven…

Let D be a finite set of positive real numbers. The distance graph G(R,D) is the graph with vertex set R (set of real numbers), and two vertices x, y are adjacent if |x-y| belongs to D. We prove that every positive integer t>1 there is a…

We prove that for every integer $k$, every finite set of points in the plane can be $k$-colored so that every half-plane that contains at least $2k-1$ points, also contains at least one point from every color class. We also show that the…

We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a translated and rotated copy of $T$ contained in $C$ or in $O$. Apart from that, we consider…