Related papers: Bounds for Pach's selection theorem and for the mi…

Let ${\mathcal D}(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove…

The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of…

Given a finite set of points $S\subset\mathbb{R}^d$, a $k$-set of $S$ is a subset $A \subset S$ of size $k$ which can be strictly separated from $S \setminus A $ by a hyperplane. Similarly, a $k$-facet of a point set $S$ in general position…

Let $Z \subseteq \proj{n}$ be a fat points scheme, and let $d(Z)$ be the minimum distance of the linear code constructed from $Z$. We show that $d(Z)$ imposes constraints (i.e., upper bounds) on some specific shifts in the graded minimal…

The \textit{$r$-dynamic choosability} of a graph $G$, written ${\rm ch}_r(G)$, is the least $k$ such that whenever each vertex is assigned a list of at least $k$ colors a proper coloring can be chosen from the lists so that every vertex $v$…

We give superexponential lower and upper bounds on the number of coloured $d$-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and $d\geq 3$ is fixed. In the special…

We say that a family ${x_i|i\in[m]}$ of vectors in a Banach space $X$ satisfies the $k$-collapsing condition if $|\sum_{i\in I}x_i|\leq 1$ for all $k$-element subsets $I\subseteq{1,2,...,m}$. Let $C(k,d)$ denote the maximum cardinality of a…

According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an $\eps$-net of size $O\left(\frac{1}{\eps}\log\frac1{\eps}\right)$. Using probabilistic techniques, Pach and Woeginger (1990)…

We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane…

We present a method for finding large fixed-size primes of the form $X^2+c$. We study the density of primes on the sets $E_c = \{N(X,c)=X^2+c,\ X \in (2\mathbb{Z}+(c-1))\}$, $c \in \mathbb{N}^*$. We describe an algorithm for generating…

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here.…

We prove a new upper bound for the minimum $d$-degree threshold for perfect matchings in $k$-uniform hypergraphs when $d<k/2$. As a consequence, this determines exact values of the threshold when $0.42k \le d < k/2$ or when $(k,d)=(12,5)$…

We derive upper and lower bounds on the gradients of Wachspress coordinates defined over any simple convex d-dimensional polytope P. The bounds are in terms of a single geometric quantity h_*, which denotes the minimum distance between a…

The main goal of the paper is to provide a quantitative lower bound greater than $1$ for the relative projection constant $\lambda(Y, X)$, where $X$ is a subspace of $\ell_{2p}^m$ space and $Y \subset X$ is an arbitrary hyperplane. As a…

Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show…

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known…

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 $d,n$…

We consider percolation on $\mathbb{Z}^d$ and on the $d$-dimensional discrete torus, in dimensions $d \ge 11$ for the nearest-neighbour model and in dimensions $d>6$ for spread-out models. For $\mathbb{Z}^d$, we employ a wide range of…

We study a double Dirichlet series of the form $\sum_{d}L(s,\chi_{d}\chi)\chi'(d)d^{-w}$, where $\chi$ and $\chi'$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to…

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…