On the finiteness of near polygons with 3 points on every line
The author deals with near \(2d\)-gons. A {near \(2d\)-gon} is a partial linear space \(\Gamma = ({\mathcal P}, {\mathcal L}, I)\) satisfying the following two conditions: (1) For every point \(p \in {\mathcal P}\) and every line \(l \in {\mathcal L}\) there exists a unique point on \(l\) with minimal distance to \(p\), where the distance is measured in the collinearity graph of \(\Gamma\). (2) The diameter of \(\Gamma\) equals \(d\). A near \(2d\)-gon is called of order \((s, t)\), if each point is incident with exactly \(t+1\) lines and if each line is incident with exactly \(s+1\) points. \(s\) and \(t\) are not necessarily finite. Note that a near quadrangle is already a generalized quadrangle. The main result of de Bruyn is as follows: Theorem. Let \(\Gamma\) be a near \(2d\)-gon of order \((2, t)\) such that any two points at distance 2 have at least two common neighbours. Then \(\Gamma\) is finite.
scientific article; zbMATH DE number 1985815