{
  "id": "2301.05830",
  "version": "v1",
  "published": "2023-01-14T06:28:13.000Z",
  "updated": "2023-01-14T06:28:13.000Z",
  "title": "Four-vertex traces of finite sets",
  "authors": [
    "Peter Frankl",
    "Jian Wang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $[n]=X_1\\cup X_2\\cup X_3$ be a partition with $\\lfloor\\frac{n}{3}\\rfloor \\leq |X_i|\\leq \\lceil\\frac{n}{3}\\rceil$ and define $\\mathcal{G}=\\{G\\subset [n]\\colon |G\\cap X_i|\\leq 1, 1\\leq i\\leq 3\\}$. It is easy to check that the trace $\\mathcal{G}_{\\mid Y}:=\\{G\\cap Y\\colon G\\in \\mathcal{G}\\}$ satisfies $|\\mathcal{G}_{\\mid Y}|\\leq 12$ for all 4-sets $Y\\subset [n]$. For $n\\geq 25$ it is proven that whenever $\\mathcal{F}\\subset 2^{[n]}$ satisfies $|\\mathcal{F}|>|\\mathcal{G}|$ then $|\\mathcal{F}_{\\mid C}|\\geq 13$ for some $C\\subset [n]$, $|C|=4$. Several further results of a similar flavor are established as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-14T06:28:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite sets",
      "four-vertex traces",
      "similar flavor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}