Quantitative axioms for finite affine planes

On Finite Point Transitive Affine Planes with Two Orbits

Finite flag-transitive affine planes with a solvable automorphism group

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associ...

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 F2q over a finite field Fq, where the formally defined squared Euclidean distance of every pair of points is a square in Fq. It turns out that integral point sets over Fq can also be characterized as affine poi...

Berlekamp's Switching Game on Finite Projective and Affine Planes

We adapt Berlekamp’s light bulb switching game to finite projective plans and finite affine planes, then find the worst arrangement of lit bulbs for planes of even and odd orders. The results are then extended from the planes to spaces of higher dimension. 1 Berlekamp’s Switching Game The original game is simple enough. 100 light bulbs arranged in 10 rows and 10 columns. Each column and row has...

Axioms of Affine Buildings

We prove equivalence of certain axiom sets for a ne buildings. Along the lines a purely combinatorial proof of the existence of a spherical building at in nity is given. As a corollary we obtain that being an a ne building is independent of the metric structure of the space.