Related papers: The maximum diameter of $d$-dimensional simplicial…
We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times…
We investigate simplicial complexes deterministically growing from a single vertex. In the first step, a vertex and an edge connecting it to the primordial vertex are added. The resulting simplicial complex has a 1-dimensional simplex and…
A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower…
The core of this note is the observation that links between circle packings of graphs and potential theory developed in \cite{BeSc01} and \cite{HS} can be extended to higher dimensions. In particular, it is shown that every limit of finite…
New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds…
We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$'s density. This generalizes a well-known classical result of Erd\H{o}s and Gallai \cite{EG59} for graphs. We use methods from…
In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the…
It is well known that a tournament (complete oriented graph) on $n$ vertices has at most ${1/4}\binom{n}{3}$ directed triangles, and that the constant 1/4 is best possible. Motivated by some geometric considerations, our aim in this paper…
We consider the hypergraph Tur\'an problem of determining $\mathrm{ex}(n, S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on $n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of $S^d$) as a…
Estimates of the fractal dimension $D$ of the set of galaxies in the universe, based on ever improving data sets, tend to settle on $D\approx 2$. This result raised a raging debate due to its glaring contradiction with astrophysical models…
Given any two vertices u, v of a random geometric graph, denote by d_E(u,v) their Euclidean distance and by d_G(u,v) their graph distance. The problem of finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of attention…
The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of…
In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the…
We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every…
We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on…
In Graph Minor III, Robertson and Seymour conjecture that the tree-width of a planar graph and that of its dual differ by at most one. We prove that given a hypergraph H on a surface of Euler genus k, the tree-width of H^* is at most the…
A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of…
We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$…
The maximal contact symmetry dimensions for scalar ODEs of order $\ge 4$ and vector ODEs of order $\ge 3$ are well known. Using a Cartan-geometric approach, we determine for these ODEs the next largest realizable (submaximal) symmetry…
An old conjecture of Bollob\'as and Scott asserts that every Eulerian directed graph with average degree $d$ contains a directed cycle of length at least $\Omega(d)$. The best known lower bound for this problem is $\Omega(d^{1/2})$ by…