  We use $p$-rank bounds on partial ovoids and the classical bounds on Ramsey
numbers to obtain various upper bounds on partial $m$-ovoids in finite polar
spaces. These bounds imply non-existence of $m$-ovoids for various new families
of polar spaces. We give a probabilistic construction of large partial
$m$-ovoids when $m$ grows linearly with the rank of the polar space.
  In the special case of the symplectic spaces over the binary field, we show
an equivalence between partial $m$-ovoids and a generalisation of the Oddtown
theorem from extremal set theory that has been studied under the name of nearly
$m$-orthogonal sets over finite fields. We give new constructions for partial
$m$-ovoids in these spaces and thus $m$-nearly orthogonal sets, for small
values of $m$. These constructions use triangle-free graphs whose complements
have low $\mathbb{F}_2$-rank and we give an asymptotic improvement over the
state of the art. We also prove new lower bounds in the recently introduced
rank-Ramsey problem for triangles vs cliques
