Ramsey numbers and extremal structures in polar spaces

TL;DR

This paper establishes new upper bounds on partial ovoids in finite polar spaces using Ramsey theory and p-rank bounds, introduces probabilistic and algebraic constructions for large partial ovoids, and connects these to extremal set theory.

Contribution

It provides novel bounds on partial ovoids, links extremal set theory to polar spaces, and offers improved constructions using graph theory and coding theory.

Findings

New upper bounds on partial ovoids in polar spaces.

Probabilistic methods for constructing large partial ovoids.

Improved constructions of partial 2-ovoids using BCH codes and graph theory.

Abstract

We use $p$-rank bounds on partial ovoids and the classical bounds on Ramsey numbers to obtain upper bounds on the size of partial $m$-ovoids in finite classical polar spaces. These bounds imply non-existence of $m$-ovoids for new infinite families of polar spaces. We also 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 prove an equivalence between partial $m$-ovoids and a generalisation of Oddtown families from extremal set theory that has been studied under the name of $m$-nearly orthogonal sets. We give a new construction for large partial $2$-ovoids in these spaces and thus $2$-nearly orthogonal sets over the binary field. This construction uses triangle-free graphs associated to certain BCH codes whose complements have low $2$-rank and it gives an asymptotic improvement over the previous best construction. We give another construction of triangle-free graphs using a binary projective cap, which has low complementary rank over the reals. This improves the bounds in the recently introduced rank-Ramsey problem and it gives better constructions of large partial $m$-ovoids for $m > 2$ in the binary symplectic space.