Related papers: A remark on sets with few distances in $\mathbb{R}…

We consider a distance-regular graph $\Gamma=(X, \mathcal R)$ called the bilinear forms graph $H_q(D,N-D)$; we assume $N>2D\geq 6$ and $q \not=2$. We show that $\Gamma$ satisfies the following strengthened version of the balanced set…

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented…

Let $S$ be a set of $n$ points in $d$-dimensional Euclidean space. Assign to each $x\in S$ an arbitrary distance $r(x)>0$. Let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\"os and Pach (1988)…

We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…

Takens' Embedding Theorem remarkably established that concatenating M previous outputs of a dynamical system into a vector (called a delay coordinate map) can be a one-to-one mapping of a low-dimensional attractor from the system state…

A combinatorial theorem on families of disjoint sub-boxes of a discrete cube, which implies that there at most $2^{d+1}-2$ neighbourly simplices in $\mathbb R^d$, is presented.

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in $\mathbb{R}^n$, we have $|A+B|^{1/n} \geq |A|^{1/n}+|B|^{1/n}$. Also, equality holds if and only if $A$ and $B$ are convex and homothetic sets in…

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…

We present some elementary ideas to prove the following Sylvester-Gallai type theorems involving incidences between points and lines in the planes over the complex numbers and quaternions. (1) Let A and B be finite sets of at least two…

We prove that if $P$ is a set of $n$ points in $\mathbb{C}^2$, then either the points in $P$ determine $\Omega(n^{1-\epsilon})$ complex distances, or $P$ is contained in a line with slope $\pm i$. If the latter occurs then each pair of…

Shnirel'man's inequality and Shnirel'man's basis theorem are fundamental results about sums of sets of positive integers in additive number theory. It is proved that these results are inherently order-theoretic and extend to partially…

We study Constraint Satisfaction Problems (CSPs) in an infinite context. We show that the dichotomy between easy and hard problems -- established already in the finite case -- presents itself as the strength of the corresponding De…

Let $B$ be a bilinear form on pairs of points in the complex plane, of the form $B(p,q) = p^TMq$, for an invertible $2\times2$ complex matrix $M$. We prove that any finite set $S$ contained in an irreducible algebraic curve $C$ of degree…

An inequality is derived for the average $t$-energy of pinned distance measures, where $0 < t < 1$. This refines Mattila's theorem on distance sets to pinned distance sets, and gives an analogue of Liu's theorem for pinned distance sets of…

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$-derivations view as the set of polynomial vector fields which possess $\mathcal{A}$ as an invariant set. We first characterize…

The wild part of Abhyankar's Inertia Conjecture for a product of certain Alternating groups is shown for any algebraically closed field of odd characteristic. For $d$ a multiple of the characteristic of the base field, a new \'etale…

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an…

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…

While numerous extensions of Banach's fixed point theorem typically offer only sufficient conditions for the existence and uniqueness of a fixed point and the convergence of iterative sequences, this study introduces a generalization…

We seek for lines of minimal distance to finitely many points in the plane. The distance between a line and a set of points is defined by the L^p-norm, 1\leq p\leq \infty, of the vector of vertical or orthogonal distances from the single…