Tight sets in finite classical polar spaces

Abstract

We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries $\text{PG}(2r+1,\,\sqrt{q})$ and generators of H(2r + 1, q), if q ≥ 81 is an odd square and i < (q2/3 − 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Δ, Δ⊥}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Δ, Δ⊥} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity ⊥ of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where $i \lt q^{5/8} / \sqrt{2} +1$ was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q2/3 − 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).

Perfect 2-Colorings of the Grassmann Graph of Planes

“…Tight sets and intriguing sets in the geometric setting cover a variety of topics of interest to finite geometers, including movoids of generalized quadrangles and Cameron-Liebler classes among others. Intriguing sets, with a focus on tight sets, in (the point graph of) various finite geometries have received a lot of attention in recent years; see for example [1,6,7,11].…”

9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

“…Known results on m-ovoids and i-tight sets are surveyed in [5,6]. See [22,26,28,34,78,71] for recent constructions of m-ovoids and i-tight sets.…”

Constructions of tight sets of the Hermitian polar space $\mc{H}(2r-1,q^2)$

“…About characterization results, some small i-tight sets of H(2r − 1, q 2 ) are characterized. Nakić and Storme [14] showed that an i-tight set in H(2r − 1, q 2 ), q ≥ 9 odd, must be a union of pairwise disjoint generators of H(2r − 1, q 2 ) and (2r − 1)-dimensional Baer subgeometries PG(2r − 1, q) when i < (q 4/3 − 1)/2. Similar results on small tight sets of H(2r, q 2 ) can be found in [6,13,14].…”

“…Nakić and Storme [14] showed that an i-tight set in H(2r − 1, q 2 ), q ≥ 9 odd, must be a union of pairwise disjoint generators of H(2r − 1, q 2 ) and (2r − 1)-dimensional Baer subgeometries PG(2r − 1, q) when i < (q 4/3 − 1)/2. Similar results on small tight sets of H(2r, q 2 ) can be found in [6,13,14]. For known constructions of tight sets in H(2r, q 2 ) we refer readers to [1,5,6].…”