Tight sets of points in the half-spin geometry related to Q+(9, q)

Let HS(9, q) denote the half-spin geometry associated with a nonsingular hyperbolic quadric Q+(9, q) of PG(9, q). Let X be a set of points of HS(9, q) and let N1 denote the total number of ordered pairs of distinct collinear points of HS(9, q) belonging to X. Using the extended Higman–Sims technique we will derive an upper and lower bound for N1 in terms of ∣X∣. Sets of points attaining these bounds are respectively called tight sets of Type I and tight sets of Type II. We provide examples of tight sets which are related to HS(7, q)-subspaces and 1- and 2-systems of Q+(9, q). We show that the size of the intersection of a tight set X of Type I and a tight set Y of Type II only depends on ∣X∣ and ∣Y∣. We characterize tight sets by means of this property.