{
  "id": "1601.07206",
  "version": "v1",
  "published": "2016-01-26T21:55:54.000Z",
  "updated": "2016-01-26T21:55:54.000Z",
  "title": "Sets in Almost General Position",
  "authors": [
    "Luka Milićević"
  ],
  "comment": "20 pages, no figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Erd\\H{o}s asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? F\\\"uredi constructed a set of $n$ points in almost general position with no more than $o(n)$ points in general position. Cardinal, T\\'oth and Wood extended this result to $\\mathbb{R}^3$, finding sets of $n$ points with no 5 on a plane whose subsets with no 4 points on a plane have size $o(n)$, and asked the question for higher dimensions: for given $n$, is it still true that the largest subset in general position we can guarantee to find has size $o(n)$? We answer their question for all $d$ and derive improved bounds for certain dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-26T21:55:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general position",
      "higher dimensions",
      "largest subset",
      "finding sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}