Embedding linear spaces with two line degrees in finite projective planes

In this paper we shall classify all finite linear spaces with line degrees n and n-k having at most n²+n+1 lines. As a consequence of this classification it follows: If n is large compared with k, then any such linear space can be embedded in a projective plane of order n–1 or n.     

Configurations in finite projective planes

We find necessary conditions that should be imposed on a number n for the existence of an arbitrary configuration in at least one finite projective plane of order n. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1345–1359, October, 1997.     

Conics in finite projective planes

There does not exist a general theory of conics in finite projective planes, because the many definitions of conics which are equivalent in desarguesian projective planes yield different types of conics in more general situations. Thus even the use of the word conic can lead to confusion, particularly in the finite case. This note is an attempt to clarify these various definitions and give as an example in a finite projective plane a von Staudt conic which is not an Ostrom conic. We conjecture that any finite projective plane admitting an Ostrom conic must be desarguesian.     

On resolvable Steiner 2-designs and maximal arcs in projective planes

A combinatorial characterization of resolvable Steiner 2-(v, k, 1) designs embeddable as maximal arcs in a projective plane of order \((v-k)/(k-1)\) is proved, and a generalization of a conjecture by Andries Brouwer (Geometries and groups, Springer, Heidelberg, 1981) is formulated.     

Subplanes of finite projective planes

We study questions of the hereditariness of the property of (, )-transitivity (the property of being a plane of translations) of a finite protective plane for its subplanes. In particular it is shown that hereditariness holds for subplanes containing the point .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1101–1102, July–August, 1991.     

Two-transitive orbits in finite projective planes

The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.     

Blocking structures in finite projective planes

Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 3^2h). Also, we show that these blocking sets of PG(2, q²), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 3^2h) gives rise to a family of blocking semiovals in the André plane as well.     

Some results on finite projective planes

This work contains a survey of some basic results on finite projective planes which were obtained by a group of mathematicians from Krasnoyarsk since 1985. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 63, Algebra-13, 1999.     

On homologies of finite projective planes

Study of finite projective planes which have a polarity preserved by many homologies. This research was supported in part by NSF Grant GP-33223.     

Sets of type (0,n) in Steiner systems and in finite projective planes

In this paper we study the sets of type (0,n)₁ in a finite projective plane and the inclusion relations between them.