The completion of the classification of the finite dense near 2d-gons, d ∈ {3,4}, with four points on each line

Abstract

We complete the classification of all finite dense near hexagons and octagons with four points on each line by filling the gaps that exist in the current (partial) classification results. The filling of these gaps will be possible after a prior investigation in which we obtain several classification and nonexistence results regarding finite near hexagons that satisfy the following three properties for certain s,t2∈N∖{0,1}: every line contains precisely s+1 points; every two points at distance 2 have either 2 or t2+1 common neighbours; if Q is a quad of order (s,t2), then Γ2(Q) does not contain lines.