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 numbers of low-dimensional spaces

New lower bounds are found for the minimum number of colors needed to color all points of a Euclidean space in such a way that any two points at a distance of 1 have different colors.     

On the chromatic numbers of small-dimensional Euclidean spaces

In this paper, we will give a short survey concerning estimates on the chromatic numbers of Euclidean spaces in small dimensions. We will also present some new important bounds.     

Borsuk's problem

The Borsuk number of a bounded set F is the smallest natural number k such that F can be represented as a union of k sets, the diameter of each of which is less than diam F. In this paper we solve the problem of finding the Borsuk number of any bounded set in an arbitrary two-dimensional normed space (the solution is given in terms of the enlargement of a set to a figure of constant width). We indicate spaces for which the solution of Borsuk's problem has the same form as in the Euclidean plane.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 621–631, November, 1977.     

On the chromatic number for a set of metric spaces

We study the problem of finding the chromatic number of a metric space with a forbidden distance. Using the linear-algebraic technique in combinatorics and convex optimization methods, we obtain a set of new estimates and observe the change of the asymptotic lower bound for the chromatic number of Euclidean space under the continuous change of the metric from l ₁ to l ₂.     

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets ${K\subset H}$ of a Hilbert space H by the metric entropy of the set K where the covering numbers ${N(K, \varepsilon)}$ of K by ${\varepsilon}$ -balls of H satisfy the Lorentz condition $$ \int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty $$ for some fixed ${0 < r, s \le \infty}$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, ?∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.     

Total chromatic numbers

We determine the total chromatic number of the join of two paths, the cartesian product of two paths, the cartesian product of a path and a cycle, the corona of two graphs and the theta graphs.     

Singular numbers of contractions in spaces with an indefinite metric and Yamamoto's theorem

A theorem of Yamamoto on singular numbers of matrices is extended to G-bounded matrices.     

Ultrametricity and metric betweenness in tangent spaces to metric spaces

The paper deals with pretangent spaces to general metric spaces. An ultrametricity criterion for pretangent spaces is found and it is closely related to the metric betweenness in the pretangent spaces.     

Embedding metric spaces into normed spaces and estimates of metric capacity

Let M^d{\cal M}^d be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of M^d{\cal M}^d as the maximal m ? \Bbb Nm \in {\Bbb N} such that every m-point metric space is isometric to some subset of M^d{\cal M}^d (with metric induced by M^d{\cal M}^d ). We obtain that the metric capacity of M^d{\cal M}^d lies in the range from 3 to ë\frac32d û+1\left\lfloor\frac{3}{2}d\right\rfloor+1 , where the lower bound is sharp for all d, and the upper bound is shown to be sharp for d ∈ {2, 3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d + 2 to ë\frac32d û+1\left\lfloor\frac{3}{2}d\right\rfloor+1 .     

Embedding metric spaces into normed spaces and estimates of metric capacity

Let be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of as the maximal such that every m-point metric space is isometric to some subset of (with metric induced by ). We obtain that the metric capacity of lies in the range from 3 to , where the lower bound is sharp for all d, and the upper bound is shown to be sharp for d ∈ {2, 3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d + 2 to . Research supported by the German Research Foundation, Project AV 85/1-1.     

Oriented hypergraphs,stability numbers and chromatic numbers

Oriented hypergraphs are defined, so that it is possible to generalize propositions characterizing the chromatic number and the stability number of a graph by means of orientations and elementary paths, to the strong and weak chromatic number and the strong and weak stability number of a hypergraph.     

Harnack's inequality for a nonlinear eigenvalue problem on metric spaces

We prove Harnack's inequality for first eigenfunctions of the p-Laplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires a weak (1,p)-Poincaré inequality. As a by-product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains.