$(r,s)$-sets from Desarguesian ovoids

Abstract

An $(r,s)$-$\text{s}et$ in $\mathrm{PG}(n,q)$ is a set of points, say $\mathcal{X}$, such that each $s$-dimensional projective subspace contains at most $r$ points of $\mathcal{X}$. We investigate $(n,n-2)$-sets and $(n-2,n-3)$-sets in $\mathrm{PG}(n,q)$, $n\le 6$. We show that the trivial upper bounds on $(n,n-2)$-sets in $\mathrm{PG}(n,q)$, $4\le n\le 6$, $(4,3)$-sets in $\mathrm{PG}(6,q)$ and $(3,2)$-sets in $\mathrm{PG}(5,q)$ are essentially sharp. A $(3,2)$-set in $\mathrm{PG}(13,q)$ of size $\frac{q^6-1}{q-1}$ is also constructed.