Embedding the linear structure of planar spaces into projective spaces (Q676716)

A triple \(\mathbb{L}: =(\mathbb{P}, {\mathcal G}, {\mathcal E})\) is called a planar space if \({\mathcal G}, {\mathcal E}\) are subsets of the power set of \(\mathbb{P}\) such that: 1. \(\forall a,b \in\mathbb{P}\), \(a\neq b:I_1G \in {\mathcal G}: a,b\in G\), if \(c\in \mathbb{P} \backslash G\), \(\exists_1 E\in {\mathcal E}: a,b,c \in E\) and if \(a,b\in F\in {\mathcal E}\) then \(G\subset F\). 2. \(\forall G\in {\mathcal G}: |G|\geq 2\), \(\forall E\in {\mathcal E}: |E |\geq 3\) and \(E\notin {\mathcal G}\), \(|{\mathcal G} |,|{\mathcal E} |\geq 2.\) \(\mathbb{L}\) is called degenerated if there is either a plane \(E\in {\mathcal E}\) with \(|\mathbb{P} \backslash E|=1\) or two lines \(A,B\in {\mathcal G}\) with \(\mathbb{P}= A\cup B\). The author considers finite nondegenerated planar spaces with \(v:= |\mathbb{P} |\), \(\pi: =|{\mathcal E} |\in \mathbb{N}\) and proves: If \(\pi\leq v+ 10^{-3} \cdot v^{5/6}\) then there exists a prime power \(q\leq \root 3\of {v} +3000^{-1} \root 6\of {v}\) such that \((\mathbb{P}, {\mathcal G})\) can be embedded into the projective space \(PG(3,q)\) and moreover that most of the planes of \({\mathcal E}\) are induced by planes of \(PG(3,q)\).