{
  "id": "2406.03043",
  "version": "v1",
  "published": "2024-06-05T08:14:41.000Z",
  "updated": "2024-06-05T08:14:41.000Z",
  "title": "Ramsey numbers and extremal structures in polar spaces",
  "authors": [
    "John Bamberg",
    "Anurag Bishnoi",
    "Ferdinand Ihringer"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "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",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-05T08:14:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ramsey numbers",
      "extremal structures",
      "orthogonal sets",
      "finite polar spaces",
      "construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}