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

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We prove that a non-empty set L of at most q⁵+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 0 or q+1 elements of L, (2) every plane 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 2q+1 elements of L, and (4) every 4-dimensional subspace of PG(n, q) is incident with at most q³-q²+4q elements of L, is necessarily the set of lines of a split Cayley hexagon H(q) naturally embedded in PG(6, q).