On the uniqueness of near polygons with three points on every line
The author presents a criterion to decide whether two near \(2d\)-gons (A) with three points per line, (B) such that every two non collinear points which are both collinear with at least one point are collinear with at least two points, and (C) containing a big geodesically closed sub-near \(2(d-1)\)-gon \({\mathcal H}\), are isomorphic (\({\mathcal H}\) is big if every point outside \({\mathcal H}\) is collinear with a (unique) point of \({\mathcal H}\)). The criterion basically says that the incidence structure consisting of the lines of some near \(2d\)-gon satisfying (A), (B) and (C) meeting \({\mathcal H}\) in one point, and the \(3\times 3\) grids which have one line in common with \({\mathcal H}\) (with natural incidence relation) determines the near \(2d\)-gon unambiguously. This is a very beautiful result, since most of the known near polygons that satisfy (A) and (B) also satisfy (C). The author applies this criterion to two specific near hexagons to give a geometric prove of their uniqueness (but the uniqueness itself was previously known).
scientific article; zbMATH DE number 1884139