{
  "id": "2106.03261",
  "version": "v1",
  "published": "2021-06-06T22:08:59.000Z",
  "updated": "2021-06-06T22:08:59.000Z",
  "title": "Which graphs can be counted in $C_4$-free graphs?",
  "authors": [
    "David Conlon",
    "Jacob Fox",
    "Benny Sudakov",
    "Yufei Zhao"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$ edges, then the density of $F$ in $G$, after a suitable normalization, is approximately at least the density of $F$ in an $\\epsilon$-regular approximation of $G$. In recent work, motivated by applications in extremal and additive combinatorics, we showed that $C_5$ has this property. Here we construct a family of graphs with the property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-06T22:08:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free graph",
      "regular approximation",
      "suitable normalization",
      "identifying graphs",
      "counting lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}