  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 $3 \leq k\leq n-2$. We give an overview of the
largest examples of these sets with size more than
$f(k,q)=\max\{3q^4+6q^3+5q^2+q+1,\theta_{k+1}+q^4+2q^3+3q^2\}$.
