Arcs in Desarguesian nets

A trivial upper bound on the size k of an arc in an r-net is $k \leq r + 1$. It has been known for about 20 years that if the r-net is Desarguesian and has odd order, then the case $k = r + 1$ cannot occur, and $k \geq r - 1$ implies that the arc is contained in a conic. In this paper, we show that actually the same must hold provided that the difference $r - k$ does not exceed $\sqrt{k/18}$. Moreover, it is proved that the same assumption ensures that the arc can be extended to an oval of the net

Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $\mathbb{F}_q$. It turns out that integral point sets over $\mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over $\mathbb{F}_q$ for $q\le 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality.Comment: 18 pages, 3 figures, 2 table

Generalizing Korchmáros-Mazzocca arcs

In this paper, we generalize the so called Korchmáros-Mazzocca arcs, that is, point sets of size $q+t$ intersecting each line in 0, 2 or t points in a finite projective plane of order q. When t is not 2 then this means that each point of the point set is incident with exactly one line meeting the point set in t points. In PG(2,p^n), we change 2 in the definition above to any integer m and describe all examples when m or t is not divisible by p. We also study mod p variants of these objects, give examples and under some conditions we prove the existence of a nucleus

Affinely regular polygons in an affine plane

In this paper we survey results about affinely regular polygons. First, the definitions and classification of affinely regular polygons are given. Then the theory of Bachmann-Schmidt is outlined. There are several classical theorems about regular polygons, some of them having analogues in finite planes, such as the Napoleon-Barlotti theorem. Such analogues, variants of classical theorems are also collected. Affinely regular polygons occur in many combinatorial problems for sets in a finite plane. Some of these results about sharply focused arcs, internal and external nuclei, complete arcs are collected. Finally, bounds on the number of chords of an affinely regular polygon through a point are discussed