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

TL;DR

This paper studies special point sets in projective spaces, establishing bounds and constructions for various dimensions and parameters, including a new large set in PG(13, q).

Contribution

It determines sharp bounds for certain (r,s)-sets in PG(n, q) and constructs a large (3, 2)-set in PG(13, q).

Findings

Sharp bounds for (n, n-2)-sets in PG(n, q) for 4 ≤ n ≤ 6.

Construction of a large (3, 2)-set in PG(13, q).

Abstract

An $(r, s)$-${\textit set}$ in ${\rm PG}(n, q)$ 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 ${\rm PG}(n, q)$, $n \le 6$. We show that the trivial upper bounds on $(n, n-2)$-sets in ${\rm PG}(n, q)$, $4 \le n \le 6$, $(4, 3)$-sets in ${\rm PG}(6, q)$ and $(3, 2)$-sets in ${\rm PG}(5, q)$ are essentially sharp. A $(3, 2)$-set in ${\rm PG}(13, q)$ of size $\frac{q^6-1}{q-1}$ is also constructed.