Near hexagons with four points on a line (Q2736574)

A near polygon is a connected partial linear space satisfying the property that for every point \(x\) and every line \(L\), there is a unique point on \(L\) nearest to \(x\). (Distance is measured in the collinearity graph \(\Gamma\).) If \(d\) is the diameter of \(\Gamma\), then the near polygon is called a near \(2d\)-gon. A near polygon is said to have order \((s,t)\) provided each line is incident with \(1 + s\) points and each point is incident with \(1+t\) lines.NEWLINENEWLINENEWLINEIn the paper under review there is a significant attempt to classify all near hexagons (i.e., \(d = 3\)) with the two properties: (i) each line is incident with 4 points \((s = 3)\); (ii) every two points at distance 2 have at least two common neighbors.NEWLINENEWLINENEWLINEThe author begins by giving ten explicit examples which are either `classical' or `glued', i.e., given by previously known general techniques. He then proceeds to show that if a near hexagon satisfies his two properties (i) and (ii), then either it is one of the ten examples or it fits one of four very specific sets of restrictions, for each of which there is no known example. For example, the largest of the open cases would be a near hexagon with order \((3,34)\) on 20608 points, with the property that each pair of points at distance 2 is contained in a unique \(4\times 4\) grid.