تقرير
Ramsey numbers and extremal structures in polar spaces
| العنوان: | Ramsey numbers and extremal structures in polar spaces |
|---|---|
| المؤلفون: | Bamberg, John, Bishnoi, Anurag, Ihringer, Ferdinand |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics |
| الوصف: | 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 Comment: 12 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2406.03043 |
| رقم الأكسشن: | edsarx.2406.03043 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |