Constructions for large spatial point-line (n sub k) configurations - Gévay, Gábor (Ars mathematica contemporanea, 2014)
Highly symmetric figures, such as regular polytopes, can serve as a scaffolding on which spatial ?$(n_k)$? point-line configurations can be built. We give several constructions using this method in dimension 3 and 4. We also explore possible constructions of point-line configurations obtained as Cartesian products of smaller ones. Using suitable powers of well-chosen configurations, we obtain infinite series of ?$(n_k)$? configurations for which both ?$n$? and ?$k$? are arbitrarily large. We also combine the method of polytopal scaffolding and the method of powers to construct further examples. Finally, we formulate an incidence statement concerning a ?$(100_4)$? configuration in 3-space derived from the product of two complete pentalaterals; it is posed as a conjecture.
Keywords: grafi; incidence statement; kombinatorika; Platonic solid; product of configurations; regular 4-polytope; spatial configuration