Related papers: Partial covers of PG(n,q)

We are interested to bound from below the number of distinct dot products determined by a finite set of points $P$ in the Euclidean plane. In this paper, we build on the work of B. Hanson, O. Roche-Newton, and S. Senger, to obtain the…

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…

Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in…

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus…

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way…

Let $X$ be a hypersurface in $\mathbb{P}^N$ with $N\geq 3$ defined over a finite field. The main result of this note is the classification, up to projective equivalence, of hypersurfaces $X$ as above without a linear component when the…

We consider the problem of covering hypersphere by a set of spherical hypercaps. This sort of problem has numerous practical applications such as error correcting codes and reverse k-nearest neighbor problem. Using the reduction of non…

D. Khavinson and G. Swiatek proved that harmonic polynomials p(z)+q(z), where p is holomorphic, q is antiholomorphic, and deg p = n > 1 = deg q, can have at most 3n-2 complex zeros. We show that this bound is sharp for all n by proving a…

We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of…

We determine all the Q-fundamental surfaces in $(p,1)$-lens spaces and $(p,2)$-lens spaces with respect to natural triangulations with $p$ tetrahedra. For general $(p,q)$-lens spaces, we give an upper bound for elements of vectors which…

In the projective planes $\mathrm{PG}(2,q)$, we collect the smallest known sizes of complete arcs for the regions \begin{align*} &\mbox{all } q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s in the interval…

A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its…

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in $\mathbb{R}^4$. Specifically, we show that an arrangement of $n$ algebraic curves determines at most $C_\epsilon…

We provide sharp bounds for the isoperimetric constants of infinite plane graphs (tessellations) with bounded vertex and face degrees. For example, if $G$ is a plane graph satisfying the inequalities $p_1 \leq \mbox{deg}\ v \leq p_2$ for $v…

In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $<B>=PG(d,2)$ with $d$ odd and $B$ is a set of $d+2$ points of $PG(d,2)$ no $d+1$ of them in the same hyperplane. As a corollary to the…

For a set $G$ of points in $\PG(m-1,q)$, let $\ex_q(G;n)$, denote the maximum size of a collection of points in $\PG(n-1,q)$ not containing a copy of $G$, up to projective equivalence. We show that \[\lim_{n\rightarrow \infty}…

Given a triple covering $X$ of genus $g$ of a general (in the sense of Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free pencils of degree $d$ which are not composed with the triple covering for any $d\ge…

Bollob\'{a}s-type theorem has received a lot of attention due to its application in graph theory. In 2015, G\'{a}bor Heged{\"u}s gave an upper bound of bollob\'{a}s-type affine subspace families for $q\neq 2$, and constructed an almost…

Let $n$ and $t$ be positive integers with $t<n$, and let $q$ be a prime power. A $\textit{partial $(t-1)$-spread}$ of ${\rm PG}(n-1,q)$ is a set of $(t-1)$-dimensional subspaces of ${\rm PG}(n-1,q)$ that are pairwise disjoint. Let…

A collection of n distinct hyperplanes L_i ={l_i=0} in P^{n-1}, the n-1-dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a quadric Q={q=0}, if each L_i is the polar…