Ramsey numbers and extremal structures in polar spaces

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 F₂-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