A Note on Near Hexagons with Lines of Size \(3\)

Abstract

We classify all finite near hexagons which satisfy the following properties for a certain \(t_2 \in \{1,2,4\}\):(i) every line is incident with precisely three points;(ii) for every point \(x\), there exists a point \(y\) at distance \(3\) from \(x\);(iii) every two points at distance \(2\) from each other have either \(1\) or \(t_2 + 1\) common neighbours;(iv) every quad is big. As a corollary, we obtain a classification of all finite near hexagons satisfying (i), (ii) and (iii) with \(t_2\) equal to \(4\).