Chromatic Number Of The Plane Research Articles

Almost-Monochromatic Sets and the Chromatic Number of the Plane

In a colouring of {mathbb {R}}^d a pair (S,s_0) with Ssubseteq {mathbb {R}}^d and with s_0in S is almost-monochromatic if Ssetminus {s_0} is monochromatic but S is not. We consider questions about finding almost-monochromatic similar copies of pairs (S,s_0) in colourings of {mathbb {R}}^d, {mathbb {Z}}^d, and of {mathbb {Q}} under some restrictions on the colouring. Among other results, we characterise those (S,s_0) with Ssubseteq {mathbb {Z}} for which every finite colouring of {mathbb {R}} without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of (S,s_0). We also show that if Ssubseteq {mathbb {Z}}^d and s_0 is outside of the convex hull of Ssetminus {s_0}, then every finite colouring of {mathbb {R}}^d without a monochromatic similar copy of {mathbb {Z}}^d contains an almost-monochromatic similar copy of (S,s_0). Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of chi ({{mathbb {R}}}^2)ge 5.

Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres

This paper is devoted to the development of algorithms for finding unit distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Hadwiger–Nelson problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, these graphs do not use the Moser spindle as the base element. The construction of 5-chromatic graphs embedded in a sphere at two values of the radius is given. Namely, the 5-chromatic unit distance graph on 372 vertices embedded into the circumsphere of an icosahedron with a unit edge length, and the 5-chromatic graph on 972 vertices embedded into the circumsphere of a great icosahedron are constructed.

Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

The unit distance graph $G^1_{R^d}$ is the infinite graph whose nodes are points in $R^d$, with an edge between two points if the Euclidean distance between these points is $1$. The 2-dimensional version $G^1_{R^2}$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G^1_{R^d}$ to closed convex subsets $X$ of $R^d$. We show that the graph $G^1_{R^d}[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to $0$, or when $r(X)\geq 1$ and the affine dimension of $X$ is at least $2$. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.

Finite $\epsilon$-unit distance graphs

In 2005, Exoo posed the following question. Fix $\epsilon$ with $0\leq\epsilon<1$. Let $G_\epsilon$ be the graph whose vertex set is the Euclidean plane, where two vertices are adjacent iff the Euclidean distance between them lies in the closed interval $[1-\epsilon,1+\epsilon]$. What is the chromatic number $\chi(G_\epsilon)$ of this graph? The case $\epsilon=0$ is precisely the classical ``chromatic number of the plane'' problem. In a 2018 preprint, de Grey shows that $\chi(G_0)\geq 5$; the proof relies heavily on machine computation. In 2016, Grytczuk et al. proved a weaker result with a human-comprehensible but nonconstructive proof: whenever $0<\epsilon<1$, we have that $\chi(G_\epsilon)\geq 5$. (This lower bound of $5$ was later improved by Currie and Eggleton to $6$.) The De Bruijn - Erd\H{o}s theorem (which relies on the axiom of choice) then guarantees the existence, for each $\epsilon$, of a finite subgraph $H_\epsilon$ of $G_\epsilon$ such that $\chi(H_\epsilon)\geq 5$. In this paper, we explicitly construct such finite graphs $H_\epsilon$. We find that the number of vertices needed to create such a graph is no more than $2\pi(15+14\epsilon^{-1})^2$. Our proof can be done by hand without the aid of a computer.

The Chromatic Number of the Plane is At Least 5: A New Proof

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

Upper Bound on the Circular Chromatic Number of the Plane

We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of the plane with $r<7$ in the case when we require disjoint arcs for points at distance belonging to the internal [0.9327,1.0673].

New upper bound for the chromatic number of a random subgraph of a distance graph

This paper is related to the classical Hadwiger-Nelson problem dealing with the chromatic numbers of distance graphs in ℝn. We consider the class consisting of the graphs G(n, 2s + 1, s) = (V (n, 2s + 1), E(n, 2s + 1, s)) defined as follows: $$V(n,2s + 1) = \{ x = (x_1 ,x_2 ,...,x_n ):x_i \in \{ 0,1\} ,x_1 + x_2 + \cdots + x_n = 2s + 1\} ,E(n,2s + 1,s) = \{ \{ x,y\} :(x,y) = s\} ,$$ where (x, y) stands for the inner product. We study the random graph G(G(n, 2s + 1, s), p) each of whose edges is taken from the set E(n, 2s+1, s) with probability p independently of the other edges. We prove a new bound for the chromatic number of such a graph.

Zero Sums on Unit Square Vertex Sets and Plane Colorings

We prove that if a real-valued function of the plane sums to zero on the four vertices of every unit square, then it must be the zero function. This fact implies a lower bound in a “coloring of the plane” problem similar to the famous Hadwiger–Nelson problem, which asks for the smallest number of colors needed to assign every point in the plane a color so that no two points of unit distance apart have the same color.

On the Chromatic Number of Subsets of the Euclidean Plane

The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is at least 4 and at most 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational plane, sets in convex position, infinite strips, and parallel lines.

The chromatic number of the plane: The bounded case

The chromatic number of the plane is the smallest number of colors needed in order to paint each point of the plane so that no two points (exactly) unit distance apart are the same color. It is known that seven colors suffice and (at least) four colors are necessary. In order to understand two and three colorings better, it is interesting to see how large a region can be and still be two or three colorable, and how complicated the colorings need to be. This paper gives tight bounds for two and three coloring regions bounded by circles, rectangles, and regular polygons. In particular, a square is 2-colorable if and only the length of a side is ⩽ 2 / 5 , and is 3-colorable if and only the length of a side is ⩽ 8 / 65 .

Axiom of choice and chromatic number of [formula omitted

In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed examples of a distance graph on the real line R and difference graphs on the real plane R 2 whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space R n , whose chromatic number is a positive integer in the Zermelo–Fraenkel-choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory.

COMPUTATIONAL GEOMETRY COLUMN 46

The old problem of determining the chromatic number of the plane is revisited.

Axiom of choice and chromatic number of the plane

Define a graph U on the set of all points of the plane R as its vertex set, with two points adjacent iff they are distance 1 apart. The graph U ought to be called unit distance plane, and its chromatic number w is called chromatic number of the plane. Finite subgraphs of U are called finite unit distance plane graphs. In 1950 the 18-year old Edward Nelson posed the problem of finding w (see the problem’s history in [Soi1]). A number of relevant results were obtained under additional restrictions on monochromatic sets (see surveys in [CFG,KW,Soi2,Soi3]). Falconer, for example, showed [F] that w is at least 5 if monochromatic sets are Lebesgue measurable. Amazingly though, the problem has withstood all assaults in the general case, leaving us with an embarrassingly wide range for w being 4, 5, 6 or 7. In their fundamental 1951 paper [EB], Erdos and de Bruijn have shown that the chromatic number of the plane is attained on some finite subgraph. This result has naturally channeled much of research in the direction of finite unit distance graphs. One limitation of the Erdos–de-Bruijn result, however, has remained a low key: they ARTICLE IN PRESS