A characterization of exterior lines of certain sets of points in PG (2, q)

Let A and B be disjoint sets of points in PG(2, q) the Desarguesian projective plane of order q, with |A|q, |B|=q+1, such that each line through a point of A meets B (just once). Then B is a line.     

On the size of a maximal partial spread

We show that a maximal partial spread inPG(3,q) is either a spread or has at most lines. This implies that it is not possible to cover all points but the points of a Baer-subspace by lines.     

Arcs in PG(n,q), MDS-codes and three fundamental problems of B. Segre — Some extensions

To each arc of PG(n, q) an algebraic hypersurface is associated. Using this tool new results on complete arcs are obtained. Since arcs and linear MDS-codes are equivalent objects, these results can be translated in terms of codes.Dedicated to Professor J. Tits for his sixtieth birthday     

Characterization of seminuclear sets in a finite projective plane

LetC be a set ofq + a points in the desarguesian projective plane of orderq, such that each point ofC is on exactly 1 tangent, and onea+ 1-secant (a>1). Then eitherq=a + 2 andC consists of the symmetric difference of two lines, with one further point removed from each line, orq=2a + 3 andC is projectively equivalent to the set of points {(0,1,s),(s, 0, 1),(1,s, 0): -s is not a square inGF(q)}.     

Semiovals contained in the union of three concurrent lines

Semiovals which are contained in the union of three concurrent lines are studied. The notion of a strong semioval is introduced, and a complete classification of these objects in PG(2,p) and PG(2,p²), p an odd prime, is given. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 491–501, 2007     

On q-analogues and stability theorems

In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.     

Covering all points except one

In many point-line geometries, to cover all points except one, more lines are needed than to cover all points. Bounds can be given by looking at the dimension of the space of functions induced by polynomials of bounded degree.     

An easier proof of the maximal arcs conjecture

It was a long-standing conjecture in finite geometry that a Desarguesian plane of odd order contains no maximal arcs. A rather inaccessible and long proof was given recently by the authors in collaboration with Mazzocca. In this paper a new observation leads to a greatly simplified proof of the conjecture.