A characterization of the natural embedding of the split Cayley hexagon H(q) in PG(6,q) by intersection numbers

In this paper, we prove that a set LL of q5+q4+q3+q2+q+1q5+q4+q3+q2+q+1 lines of PG(6,q) with the properties that (1) every point of PG(6,q) is incident with either 0 or q+1q+1 elements of LL, (2) every plane of PG(6,q) is incident with either 0, 1 or q+1q+1 elements of LL, (3) every solid of PG(6,q) is incident with either 0, 1, q+1q+1 or 2q+12q+1 elements of LL, and (4) every hyperplane of PG(6,q) is incident with at most q3+3q2+3qq3+3q2+3q members of LL, is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon H(q) in PG(6,q).