{
  "id": "1912.11917",
  "version": "v1",
  "published": "2019-12-26T19:00:13.000Z",
  "updated": "2019-12-26T19:00:13.000Z",
  "title": "Stability theorems for the feasible region of cancellative hypergraphs",
  "authors": [
    "Xizhi Liu"
  ],
  "comment": "30 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A hypergraph is cancellative if it does not contain three sets $A,B,C$ such that the symmetric difference of $A$ and $B$ is contained in $C$. We show that for every $r\\ge 3$ a cancellative $r$-graph $\\mathcal{H}$ has a stability property whenever the sizes of $\\mathcal{H}$ and the shadow of $\\mathcal{H}$ satisfy certain inequalities. In particular, our result for $r=3$ generalizes a stability theorem of Keevash and Mubayi and it shows that for every $k\\equiv 1$ or $3$ (mod $6$) a $3$-graph $\\mathcal{H}$ is structurally close to a balanced blow up of a Steiner triple system on $k$ vertices whenever the shadow density of $\\mathcal{H}$ is close to $(k-1)/k$ and the edge density of $\\mathcal{H}$ is close to $(k-1)/k^2$. Our result for $r \\ge 3$ extends a stability theorem of Keevash about the Kruskal-Katona theorem to cancellative hypergraphs, and also addresses an old conjecture of Bollob\\'{a}s about the maximum size of a cancellative $r$-graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-26T19:00:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stability theorem",
      "cancellative hypergraphs",
      "feasible region",
      "steiner triple system",
      "symmetric difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}