Forbidden Distances Research Articles

Chromatic number of a line with geometric progressions of forbidden distances and the complexity of recognizing distance graphs

Chromatic number of a line with geometric progressions of forbidden distances and the complexity of recognizing distance graphs

The chromatic number of the space

The chromatic number of the space with the metric with forbidden distances is studied. It is defined as the minimum number of colours needed to colour all points in the space in such a way that no two points lying at a forbidden distance from each other in the metric have the same colour. The asymptotic growth of the chromatic numbers as is estimated. The instrument of study is the linear algebra method, which reduces the problem of estimating chromatic numbers to another convex extremum problem. A numerical solution of this problem makes it possible to derive sharp estimates for the constants present in the bases of the lower asymptotic estimates for the chromatic numbers of multidimensional real spaces with several forbidden distances. The estimates obtained are optimal within the framework of the method proposed. Bibliography: 27 titles.

Chromatic Numbers of Distance Graphs with Several Forbidden Distances and without Cliques of a Given Size

We consider distance graphs with k forbidden distances in an n-dimensional space with the p-norm that do not contain cliques of a fixed size. Using a probabilistic construction, we present graphs of this kind with chromatic number at least (Bk) Cn , where B and C are constants.

Chromatic Number with Several Forbidden Distances in the Space with the ℓ q -Metric

We study the chromatic number \( \overline{\chi}\left(X;\rho; k\right) \) of a metric space X with a metric ρ and k forbidden distances. We obtain an estimate of the form \( \overline{\chi}\left({\mathbb{R}}^n;\rho; k\right)\ge {(Bk)}^{Cn} \) for cases where the metric ρ on the set ℝ n is generated by the l q -norm.

Frankl-Rödl-type theorems for codes and permutations

We give a new proof of the Frankl-R\"odl theorem on forbidden intersections, via the probabilistic method of dependent random choice. Our method extends to codes with forbidden distances, where over large alphabets our bound is significantly better than that obtained by Frankl and R\"odl. We also apply our bound to a question of Ellis on sets of permutations with forbidden distances, and to establish a weak form of a conjecture of Alon, Shpilka and Umans on sunflowers.

Estimate for the chromatic number of Euclidean space with several forbidden distances

Estimate for the chromatic number of Euclidean space with several forbidden distances

Estimate for the Chromatic Number of Euclidean Space with Several Forbidden Distances

Estimate for the Chromatic Number of Euclidean Space with Several Forbidden Distances

New Lower Bound for the Chromatic Number of a Rational Space with One and Two Forbidden Distances

В работе получена новая нижняя оценка хроматического числа $\chi({\mathbb Q}^n)$ пространства ${\mathbb Q}^n$. Библиография: 14 названий.

New lower bound for the chromatic number of a rational space with one and two forbidden distances

A new lower bound for the chromatic number χ(ℚ n ) of the space ℚ n is obtained.

On the chromatic number of Euclidean space with two forbidden distances

On the chromatic number of Euclidean space with two forbidden distances

The chromatic number of ℝ n with a set of forbidden distances

The chromatic number of ℝ n with a set of forbidden distances

Estimating the chromatic numbers of Euclidean space by convex minimization methods

The chromatic numbers of the Euclidean space with forbidden distances are investigated (that is, the minimum numbers of colours necessary to colour all points in so that no two points of the same colour lie at a forbidden distance from each other). Estimates for the growth exponents of the chromatic numbers as are obtained. The so-called linear algebra method which has been developed is used for this. It reduces the problem of estimating the chromatic numbers to an extremal problem. To solve this latter problem a fundamentally new approach is used, which is based on the theory of convex extremal problems and convex analysis. This allows the required estimates to be found for any . For these estimates are found explicitly; they are the best possible ones in the framework of the method mentioned above.Bibliography: 18 titles.

Chromatic numbers of real and rational spaces with real or rational forbidden distances

Several important aspects of the Nelson-Erdős-Hadwiger classical problem of combinatorial geometry are considered. In particular, new lower bounds are obtained for the chromatic numbers of the spaces and with two, three or four forbidden distances.Bibliography: 28 titles.

On the chromatic number of Euclidean space and the Borsuk problem

The present paper is devoted to two problems: the problem of the chromatic number of a metric space going back to Nelson, Erdős, and Hadwiger and the Borsuk problem of the partition of a bounded set of positive diameter into part of smaller diameter. By the chromatic number χ(X, ρ,A) of a metric space X with metric ρ for a set of forbidden distances A we mean the minimal number of colors in which one can paint X so that no two points of one color are not at a distance (in the metric ρ) belonging to A. In what follows, we shall consider the case (X, ρ) = (Rn, | · |2), where | x− y |2 is the standard Euclidean metric. By the Borsuk number f(n) we mean the minimal number of parts into which one can divide an arbitrary bounded set of positive diameter so that the diameter of each part will be less than that of the original set. The first problem dates back to the middle of the last century, while the second one, to the thirties. In the meantime, new and diverse results were obtained [1], [2] for the chromatic number and the Borsuk number. We are concerned with asymptotic lower bounds for the chromatic number χ(Rn, | · |2, {1}) and the Borsuk number as n → ∞. Now it is known that (see, respectively, [3], [4], and [2], [5]) (1.239 . . . + o(1)) ≤ χ(R, | · |2, {1}) ≤ (3 + o(1)), (1.225 . . . + o(1)) √ n ≤ f(n) ≤ (1.224 . . . + o(1)).

On the Chromatic Numbers of and with Intervals of Forbidden Distances

In this paper, we discuss new upper bounds for the chromatic numbers of R2 and R3 with intervals of forbidden distances.

Colorings of the space ℝn with several forbidden distances

The paper is concerned with the classical problem concerning the chromatic number of a metric space, i.e., the minimal number of colors required to color all points in the space so that the distance (the value of the metric) between points of the same color does not belong to a given set of positive real numbers (the set of forbidden distances). New bounds for the chromatic number are obtained for the case in which the space is ℝn with a metric generated by some norm (in particular, lp) and the set of forbidden distances either is finite or forms a lacunary sequence.

On the chromatic number of a space with several forbidden distances

On the chromatic number of a space with several forbidden distances

On the chromatic numbers of metric spaces with few forbidden distances

In this paper we will present some of our recent results concerning the chromatic numbers of different metric spaces.

On the chromatic number of a space with two forbidden distances

On the chromatic number of a space with two forbidden distances

FORBIDDEN DISTANCES IN THE RATIONALS AND THE REALS

Our main aim in this paper is to show that there is a partition of the reals into finitely many classes with ‘many’ forbidden distances, in the following sense: for each positive real x , there is a natural number n such that no two points in the same class are at distance $x/n$ . In fact, more generally, given any infinite set $\{ c_n:n of positive rationals, there is a partition of the reals into three classes such that for each positive real x , there is some n such that no two points in the same class are at distance $c_nx$ . This result is motivated by some questions in partition regularity.