2024/06/05 by John Bamberg, Anurag Bishnoi, Bamberg, John +4
Mathematics · #05D10 #05E30 #51E12 #51E99 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2406.03043
openalex publication_date 2024/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.