A characterization of the natural embedding of the split Cayley hexagon in PG(6,q) by intersection numbers in finite projective spaces of arbitrary dimension

We prove that a non-empty set LL of at most q5+q4+q3+q2+q+1q5+q4+q3+q2+q+1 lines of PG(n,q) with the properties that (1) every point of PG(n,q) is incident with either 00 or q+1q+1 elements of LL, (2) every plane of PG(n,q) is incident with either 00, 11 or q+1q+1 elements of LL, (3) every solid of PG(n,q) is incident with either 00, 11, q+1q+1 or 2q+12q+1 elements of LL, and (4) every four-dimensional subspace of PG(n,q) is incident with at most q3−q2+4qq3−q2+4q elements of LL is necessarily the set of lines of a split Cayley hexagon H(q) naturally embedded in PG(6,q).