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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 either0 or q+1 elements of L, (2) every plane plane of PG(n, q) is incident witheither 0, 1 or q+1 elements of L, (3) every solid of PG(n, q) is incident witheither 0, 1, q+1 or 2q+1 elements of L, and (4) every 4-dimensional subspace ofPG(n, q) is incident with at most q^3-q^2+4q elements of L, is necessarily theset of lines of a split Cayley hexagon H(q) naturally embedded in PG(6, q).

discrete mathematics

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