A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers in finite projective spaces of arbitrary dimension

We prove that a non-empty set L of at most q 5 + q 4 + q 3 + q 2 + q + 1 lines of PG ( n , q ) with the properties that (1) every point of PG ( n , q ) is incident with either 0 or q + 1 elements of L , (2) every plane of PG ( n , q ) is incident with either 0 , 1 or q + 1 elements of L , (3) every solid of PG ( n , q ) is incident with either 0 , 1 , q + 1 or 2 q + 1 elements of L , and (4) every four-dimensional subspace of PG ( n , q ) is incident with at most q 3 − q 2 + 4 q elements of L is necessarily the set of lines of a split Cayley hexagon H ( q ) naturally embedded in PG ( 6 , q ) .