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In the period 1994-1999 the author wrote a series of three papers on generalized quadrangles of order (s , s 2). In this Part IV we classify all finite translation generalized quadrangles of order (s , s 2) having a kernel of size at least 3, containing a regular line not incident with the translation point. There are several applications on generalized quadrangles of order (s , s 2) having at least 2 translation points, on Moufang quadrangles, and concerning the theorem of Fong and Seitz classifying all groups with a BN -pair of rank 2. [ABSTRACT FROM AUTHOR]

Let H be a hyperbolic quadric in P G ( 3 , q ) , where q is a prime power. Let E (respectively, T ) denote the set of all lines of P G ( 3 , q ) which are external (respectively, tangent) to H . We characterize the minimum size blocking sets in P G ( 3 , q ) , q ≠ 2 , with respect to the line set E ∪ T . We also give an alternate proof characterizing the minimum size blocking sets in P G ( 3 , q ) with respect to the line set E for all odd q . [ABSTRACT FROM AUTHOR]

Two combinatorial characterizations of the Hermitian surface of the finite 3-dimensional projective space PG ( 3 , q 2 ) in terms of quasi-Hermitian varieties are given. [ABSTRACT FROM AUTHOR]

Abstract: An book is a collection of -subspaces in called pages, which cover the whole projective space and intersect in a common -subspace called the spine such that any point outside the spine is in exactly one page. An book -spread is a -spread in for which there exists an book, such that the points of each page of this book and hence the points of the spine are partitioned by -subspaces of the -spread. We commence by showing that an book -spread exists if and only if the following three conditions hold: In general the number of different kinds of book -spreads is a tiny proportion of the number of different kinds of -spreads in . In the rest of this paper we present computer-aided classification results for certain types of book 1-spreads. [Copyright &y& Elsevier]

Abstract: We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a non-split link for every signed embedding. With one trivial exception, these graphs are derived from different signings of the seven Petersen family graphs. [Copyright &y& Elsevier]

Abstract: In this article, the lines not meeting a hyperoval cone in PG(3, ), even, are characterized by their intersection properties with points and planes. [Copyright &y& Elsevier]

Abstract: The Gale transform is an involution on sets of points in a projective space. It plays a crucial role in several different subjects, such as algebraic geometry, optimization, coding theory, etc. Sometimes, in the literature, the Gale transform is used implicitly, without mentioning it as such. In the present paper, we give a short account of the algebraic and geometrical implications that the Gale transform has when it is applied to sets of points in finite projective spaces. [Copyright &y& Elsevier]

Abstract: In this paper linear sets of finite projective spaces are studied and the “dual” of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides “old” results, new ones are proven and some open questions are discussed. [Copyright &y& Elsevier]

In n -dimensional affine space over the field Z / 3 Z , a cap is given by a set of points no three of which are in a line, and the cap set problem asks for the largest possible size of an arbitrary cap. The solution to the cap set problem is known for n at most 6. In this paper, we define and apply standard diagrams. These pictures interpret a well-known technique for solving the cap set problem in a new way, allowing conclusions to be derived more easily and intuitively than before. We use standard diagrams to find caps in dimensions up to and including 4 systematically. We prove the apparently new result that in dimension 4, up to isomorphism there are exactly 20 size-18 caps, which we give explicitly. This article is the first of a series. In later articles, we plan to use the methods and results of this paper to investigate dimensions 5 and higher. The eventual goal is to solve the cap set problem in dimension 7, the first unsolved case. [ABSTRACT FROM AUTHOR]

Consider a quadratic cone K in the 3-dimensional projective space PG (3 , q) over a finite field of order q , where q is a prime power. Let E (respectively, T , S) denote the set of all lines of PG (3 , q) that are external (respectively, tangent, secant) with respect to K. We characterize the minimum size blocking sets in PG (3 , q) with respect to the line set A , where A is one of E , T , S , E ∪ T , E ∪ S and T ∪ S. [ABSTRACT FROM AUTHOR]

We give a combinatorial characterization of the family of lines of PG (3 , q) , q ≥ 7 odd, which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with the points and planes of PG (3 , q). [ABSTRACT FROM AUTHOR]