We prove that there are no semi-finite generalized hexagons with   q+1     q  +  1        points on each line containing the known generalized hexagons of order   q     q        as full subgeometries when   q     q        is equal to   3     3        or   4     4       , thus contributing to the existence problem of semi-finite generalized polygons posed by Tits. The case when   q     q        is equal to   2     2        was treated by us in an earlier work, for which we give an alternate proof. For the split Cayley hexagon of order   4     4        we obtain the stronger result that it cannot be contained as a proper full subgeometry in any generalized hexagon.
['Anurag Bishnoi', 'Bart De Bruyn']
On generalized hexagons of order $(3, t)$ and $(4, t)$ containing a subhexagon