Blocking the k-Holes of Point Sets in the Plane

Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a k-hole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the k-holes of P if any k-hole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of k-hole blocking sets, with emphasis in the case k=5

Blocking the k-holes of point sets in the plane

Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a k-hole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the k-holes of P if any k-hole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of k-hole blocking sets, with emphasis in the case k=5Peer ReviewedPostprint (author's final draft

The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems

Partial covers of PG(n,q)

AbstractIn this paper, we show that a set of q+a hyperplanes, q>13, a≤(q−10)/4, that does not cover PG(n,q), does not cover at least qn−1−aqn−2 points, and show that this lower bound is sharp. If the number of non-covered points is at most qn−1, then we show that all non-covered points are contained in one hyperplane. Finally, using a recent result of Blokhuis, Brouwer and Szőnyi [8], we remark that the bound on a for which these results are valid can be improved to a<(q−2)/3 and that this upper bound on a is sharp

Sets of generators blocking all generators in finite classical polar spaces

We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces Q(2n,q), Q-(2n+1,q) and H(2n,q^2), in terms of cones with vertex a subspace contained in the polar space and with base a generator blocking set in a polar space of rank 2.Comment: accepted for J. Comb. Theory

Compatible 4-Holes in Point Sets

Counting interior-disjoint empty convex polygons in a point set is a typical Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let $P$ be a set of $n$ points in the plane and in general position. A subset $Q$ of $P$, with four points, is called a $4$-hole in $P$ if $Q$ is in convex position and its convex hull does not contain any point of $P$ in its interior. Two 4-holes in $P$ are compatible if their interiors are disjoint. We show that $P$ contains at least $\lfloor 5n/11\rfloor {-} 1$ pairwise compatible 4-holes. This improves the lower bound of $2\lfloor(n-2)/5\rfloor$ which is implied by a result of Sakai and Urrutia (2007).Comment: 17 page

Dominating sets in projective planes

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 dominating set in a projective plane of order $q>81$ is smaller than $2q+2[\sqrt{q}]+2$ (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most $2q+\sqrt{q}+1$. In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines.Comment: 19 page

Partial ovoids and partial spreads in finite classical polar spaces

We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]