Quasi-configurations - Bokowski, Jürgen (Ars mathematica contemporanea, 2016)
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 ?$(37_4)$? and ?$(43_4)$? configurations. The existence problem of finding ?$(22_4)$?, ?$(23_4)$?, and ?$(26_4)$? configurations remains open.
Keywords: ?$(n_k)$? configurations; incidenčne strukture točk in premic; konfiguracije; point-line incidence structure; projective arrangements; projektivne razporeditve