Quasi-configurations: building blocks for point-line configurations

Jürgen Bokowski; Vincent Pilaud

doi:10.26493/1855-3974.642.bbb

paper

Quasi-configurations: building blocks for point-line configurations

arXiv:1403.7939 · doi:10.26493/1855-3974.642.bbb

Abstract

We study point-line incidence structures and their properties in the projective plane. Our motivation is the problem of the existence of $(n_{4})$ configurations, still open for few remaining values of $n$ . Our approach is based on quasi-configurations: point-line incidence structures where each point is incident to at least $3$ lines and each line is incident to at least $3$ points. We investigate the existence problem for these quasi-configurations, with a particular attention to $3∣4$ -configurations where each element is $3$ - or $4$ -valent. We use these quasi-configurations to construct the first $(3 7_{4})$ and $(4 3_{4})$ configurations. The existence problem of finding $(2 2_{4})$ , $(2 3_{4})$ , and $(2 6_{4})$ configurations remains open.

12 pages, 9 figures