Related papers: Borsuk's Problem in Metric Spaces

Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdos-Ko-Rado from 1961 says that any such system has size at most ${n-1 \choose k-1}$. A natural question, which…

We classify generalized distance-squared mappings of $\mathbb{R}^{n+1}$ into $\mathbb{R}^{2n+1}$ ($n\ge 1$) having generic central points. Moreover, we show that there does not exist a universal bad set $\Sigma\subset…

In the paper we are dealing with metric measure spaces of diameter at most one and of total measure one. Gromov introduced the sampling compactification of the set of these spaces. He asked whether the metric measure space invariants extend…

In short geometrization conjecture of W.\,Thurston (finally proved by G.~Perelman) says that any oriented $3$-manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types. In the seminal…

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than $k$ can be always mapped onto a $k$-dimensional cube by a Lipschitz map. We also show that…

The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u,w} if the…

A finite set of the Euclidean space is called an $s$-distance set provided the number of Euclidean distances in the set is $s$. Determining the largest possible $s$-distance set for the Euclidean space of a given dimension is challenging.…

A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|<1/3. The bound 1/3 is optimal. This result has been…

S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a…

Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…

For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every…

The Dushnik-Miller dimension of a partially-ordered set $P$ is the smallest $d$ such that one can embed $P$ into a product of $d$ linear orders. We prove that the dimension of the divisibility order on the interval $\{1, \dotsc, n\}$, is…

This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study…

For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on…

The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$…

Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the…

In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of $n$…

We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two subsets E and K of d-dimensional Euclidean space.

The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and…

The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length $2(n+1)$ with exactly $n+1$ many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the…