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 $4$.