arXiv subjects
Publications and source records attributed to G. A. Strukov.
We prove that for an arbitrary $\varepsilon > 0$ holds \[ χ(\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] where $χ(M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of pairs of monochromatic points at the distance 1 apart.
\emph{A maximal distance minimizer} for a given compact set $M \subset \mathbb{R}^2$ and some given $r > 0$ is a set having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $Σ\subset \mathbb{R}^2$ satisfying the inequality \[ \max_{y\in M} dist (y, Σ) \leq r. \] This paper deals with the set of maximal distance minimizers for a rectangle $M$ and small enough $r$.
Fix a compact $M \subset \mathbb{R}^2$ and $r>0$. A minimizer of the maximal distance functional is a connected set $Σ$ of the minimal length, such that \[ max_{y \in M} dist(y,Σ) \leq r. \] The problem of finding maximal distance minimizers is connected to the Steiner tree problem. In this paper we consider the case of a convex closed curve $M$, with the minimal radius of curvature greater than $r$ (it implies that $M$ is smooth). The first part is devoted to statements on structure of $Σ$: we show that the closure of an arbitrary connected component of $B_r(M) \cap Σ$ is a local Steiner tree which connects no more than five vertices. In the second part we "derive in the picture". Assume that the left and right neighborhoods of $y \in M$ are contained in $r$-neighborhoods of different points $x_1$, $x_2 \in Σ$. We write conditions on the behavior of $Σ$ in the neighborhoods of $x_1$ and $x_2$ under the assumption by moving $y$ along $M$.