arXiv subjects
Publications and source records attributed to Arsenii Sagdeev.
In 2008, Schmidt and Tuller stated a conjecture concerning optimal packing and covering of integers by translates of a given three-point set. In this note, we confirm their conjecture and relate it to several other problems in combinatorics.
We solve two related extremal-geometric questions in the $n-$dimensional space $\mathbb{R}^n_{\infty}$ equipped with the maximum metric. First, we prove that the maximum size of a right-equidistant sequence of points in $\mathbb{R}^n_{\infty}$ equals $2^{n+1}-1$. A sequence is right-equidistant if each of the points is at the same distance from all the succeeding points. Second, we prove that the maximum number of points in $\mathbb{R}^n_{\infty}$ with pairwise odd distances equals $2^n$. We also obtain partial results for both questions in the $n-$dimensional space $\mathbb{R}^n_1$ with the Manhattan distance.
For two metric spaces $\mathbb X$ and $\mathcal Y$, the chromatic number $χ(\mathbb X;\mathcal Y)$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest $k$ such that there is a coloring of the points of $\mathbb X$ with $k$ colors and no monochromatic copy of $\mathcal Y$. In this paper, we show that for each finite metric space $\mathcal{M}$ that contains at least two points the value $χ\left(\mathbb R^n_\infty; \mathcal M \right)$ grows exponentially with $n$. We also provide explicit lower and upper bounds for some special $\mathcal M$.