arXiv subjects
Publications and source records attributed to Esmeralda Nastase.
Let $n$ and $t$ be positive integers with $t (q^r-1)/(q-1)$, then the maximum size, i.e., cardinality, of a partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ is $(q^n-q^{t+r})/(q^t-1)+1$. This essentially settles a main open problem in this area. Prior to this result, this maximum size was only known for $r\in\{0,1\}$ and for $r=q=2$.
Let $n$ and $t$ be positive integers with $t 2$. The exact value of the maximum size partial $(t-1)$-spread has been recently determined for $t>θ_r$ by the authors of this paper (see Năstase-Sissokho [21]).
A subspace partition $Π$ of $V=V(n,q)$ is a collection of subspaces of $V$ such that each 1-dimensional subspace of $V$ is in exactly one subspace of $Π$. The size of $Π$ is the number of its subspaces. Let $σ_q(n,t)$ denote the minimum size of a subspace partition of $V$ in which the largest subspace has dimension $t$, and let $ρ_q(n,t)$ denote the maximum size of a subspace partition of $V$ in which the smallest subspace has dimension $t$. In this paper, we determine the values of $σ_q(n,t)$ and $ρ_q(n,t)$ for all positive integers $n$ and $t$. Furthermore, we prove that if $n\geq 2t$, then the minimum size of a maximal partial $t$-spread in $V(n+t-1,q)$ is $σ_q(n,t)$.