Maximal sets of $k$-spaces pairwise intersecting in at least a $(k-2)$-space

Abstract

In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3\le k\le n-2$. We give an overview of the largest examples of these sets with size more than $f(k,q)=\max \{3q^4+6q^3+5q^2+q+1,{\theta }_{k+1}+q^4+2q^3+3q^2\}$.