{
  "id": "2309.01857",
  "version": "v1",
  "published": "2023-09-04T23:16:38.000Z",
  "updated": "2023-09-04T23:16:38.000Z",
  "title": "On non-degenerate Turán problems for expansion",
  "authors": [
    "Dániel Gerbner"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The $r$-uniform expansion $F^{(r)+}$ of a graph $F$ is obtained by enlarging each edge with $r-2$ new vertices such that altogether we use $(r-2)|E(F)|$ new vertices. Two simple lower bounds on the largest number $\\mathrm{ex}_r(n,F^{(r)+})$ of $r$-edges in $F^{(r)+}$-free $r$-graphs are $\\Omega(n^r)$ (in the case $F$ is not a star) and $\\mathrm{ex}(n,K_r,F)$, which is the largest number of $r$-cliques in $n$-vertex $F$-free graphs. We prove that $\\mathrm{ex}_r(n,F^{(r)+})=\\mathrm{ex}(n,K_r,F)+O(n^r)$. The proof comes with a structure theorem that we use to determine $\\ex_r(n,F^{(r)+})$ exactly for some graphs $F$, every $r\\chi(F)$ and sufficiently large $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-04T23:16:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-degenerate turán problems",
      "largest number",
      "simple lower bounds",
      "proof comes",
      "free graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}