Trivial upper bounds on (n,n-2)-sets in PG(n,q) for n=4,5,6 and related cases are shown to be essentially sharp, with a (3,2)-set of size (q^6-1)/(q-1) constructed in PG(13,q) from Desarguesian ovoids.
Cossidente, On twisted tensor product group embeddings and the spin representation of symplectic groups: the caseqodd,ISRN Geom.2011, 694605 (2011), 6 pp
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$(r,s)$-sets from Desarguesian ovoids
Trivial upper bounds on (n,n-2)-sets in PG(n,q) for n=4,5,6 and related cases are shown to be essentially sharp, with a (3,2)-set of size (q^6-1)/(q-1) constructed in PG(13,q) from Desarguesian ovoids.