A Combinatorial Characterization of Extremal Generalized Hexagons

A finite generalized $2d$-gon of order $(s,t)$ with $d \in \{ 2,3,4 \}$ and $s \not= 1$ is called extremal if $t$ attains its maximal possible value $s^{e_d}$, where $e_2=e_4=2$ and $e_3=3$. The problem of finding combinatorial conditions that are both necessary and sufficient for a finite generalized $2d$-gon of order $(s,t)$ to be extremal has so far only be solved for the generalized quadrangles. In this paper, we obtain a solution for the generalized hexagons. We also obtain a related combinatorial characterization for extremal regular near hexagons.