Related papers: Avoiding secants of given size in finite projectiv…

An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J.…

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a…

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…

Let Q_K=(Q,<_Q)$ be a strongly K-dense linear order of size K for a suitable cardinal K. We prove, for all integers m > 1 that there is a finite value t_m^+ such that the set of all m-tuples from Q can be divided into t_m^+ many classes,…

It is conjectured that the question of the existence of projective planes whose order is not a power of prime is intimately linked with the problem whether there exists a set of d+1 mutually unbiased bases in a d-dimensional Hilbert space…

A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q…

Let $S$ be a set of $n$ points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of $S$ is less than $Kn^2$ for some $K=o(n^{\frac{1}{7}})$…

We prove that a typical compact set does not contain any similar copy of a given pattern. We also prove that a typical compact set of $[0,1]^{d} (d\geq 2)$ intersects any $(d-1)$-dimensional plane in at most $d$ points. We study the…

Given $E \subset \mathbb{F}_q^d$, we show that certain configurations occur frequently when $E$ is of sufficiently large cardinality. Specifically, we show that we achieve the statistically number of $k$-stars $\displaystyle\left|\left\{(x,…

Let $q$ be a fixed odd prime. We show that a finite subset $B$ of integers, not containing any perfect $q^{th}$ power, contains a $q^{th}$ power modulo almost every prime if and only if $B$ corresponds to a blocking set (with respect to…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…

In this paper, we study prime order automorphisms of generalized quadrangles. We show that, if $\mathcal{Q}$ is a thick generalized quadrangle of order $(s,t)$, where $s > t$ and $s+1$ is prime, and $\mathcal{Q}$ has an automorphism of…

A family of sets has the $(p, q)$ property if among any $p$ members of it some $q$ intersect. It is shown that if a finite family of compact convex sets in $\R^2$ has the $(p+1,2)$ property then it is pierced by $\lfloor \frac{p}{2} \rfloor…

Working over the field of order 2 we consider those complete caps (maximal sets of points with no three collinear) which are disjoint from some codimension 2 subspace of projective space. We derive restrictive conditions which such a cap…

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with eventually different maximum fields of…

Let $P$ be a set of $n$ points in the plane, and let $\mathcal C$ be a collection of $n$ simple $k$-intersecting curves, meaning that every two distinct curves of $\mathcal C$ meet in at most $k$ points. A classical theorem of Pach and…

Let $\Pi_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound…

We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general…

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such…

Two sets $A$ and $B$ of points in the plane are \emph{mutually avoiding} if no line generated by any two points in $A$ intersects the convex hull of $B$, and vice versa. In 1994, Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and…