{
  "id": "2006.04848",
  "version": "v1",
  "published": "2020-06-08T18:09:55.000Z",
  "updated": "2020-06-08T18:09:55.000Z",
  "title": "A new stability theorem for the expansion of cliques",
  "authors": [
    "Xizhi Liu",
    "Sayan Mukherjee"
  ],
  "comment": "10 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\ell \\ge r\\ge 3$. The $r$-graph $H_{\\ell+1}^{r}$ is the hypergraph obtained from $K_{\\ell+1}$ by adding a set of $r-2$ new vertices to each edge. Using a stability result for $H_{\\ell+1}^{r}$, Pikhurko determined ${\\rm ex}(n,H_{\\ell+1}^{r})$ for sufficiently large $n$. We prove a new type of stability theorem for $H_{\\ell+1}^{r}$ that goes beyond this result, and determine the structure of $H_{\\ell+1}^{r}$-free hypergraphs $\\mathcal{H}$ that satisfy a certain inequality involving the sizes of $\\mathcal{H}$ and its shadow $\\partial\\mathcal{H}$. Our result can be viewed as an extension of a stability theorem of Keevash about the Kruskal-Katona theorem to $H_{\\ell+1}^{r}$-free hypergraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-08T18:09:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stability theorem",
      "free hypergraphs",
      "kruskal-katona theorem",
      "stability result",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}