{
  "id": "2412.04066",
  "version": "v1",
  "published": "2024-12-05T11:00:23.000Z",
  "updated": "2024-12-05T11:00:23.000Z",
  "title": "A note on infinite versions of $(p,q)$-theorems",
  "authors": [
    "Attila Jung",
    "Dömötör Pálvölgyi"
  ],
  "comment": "A previous version of this work appeared as part of arXiv:2311.15646 (v1 and v2). The main theorem is included there, but the current version provides a clearer presentation and additional applications",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that fractional Helly and $(p,q)$-theorems imply $(\\aleph_0,q)$-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier $(\\aleph_0,q)$-theorems about geometric hypergraphs that were proved recently. Some of the corollaries are new results, for example, we prove that if $\\mathcal{F}$ is an infinite family of convex compact sets in $\\mathbb{R}^d$ and among every $\\aleph_0$ of the sets some $d+1$ contain a point in their intersection with integer coordinates, then all the members of $\\mathcal{F}$ can be hit with finitely many points with integer coordinates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-05T11:00:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A35"
    ],
    "keywords": [
      "infinite versions",
      "integer coordinates",
      "convex compact sets",
      "geometric hypergraphs",
      "fractional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}