{
  "id": "2102.02103",
  "version": "v1",
  "published": "2021-02-03T15:17:31.000Z",
  "updated": "2021-02-03T15:17:31.000Z",
  "title": "Hypergraphs with many extremal configurations",
  "authors": [
    "Xizhi Liu",
    "Dhruv Mubayi",
    "Christian Reiher"
  ],
  "comment": "32 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For every positive integer $t$ we construct a finite family of triple systems ${\\mathcal M}_t$, determine its Tur\\'{a}n number, and show that there are $t$ extremal ${\\mathcal M}_t$-free configurations that are far from each other in edit-distance. We also prove a strong stability theorem: every ${\\mathcal M}_t$-free triple system whose size is close to the maximum size is a subgraph of one of these $t$ extremal configurations after removing a small proportion of vertices. This is the first stability theorem for a hypergraph problem with an arbitrary (finite) number of extremal configurations. Moreover, the extremal hypergraphs have very different shadow sizes (unlike the case of the famous Tur\\'an tetrahedron conjecture). Hence a corollary of our main result is that the boundary of the feasible region of ${\\mathcal M}_t$ has exactly $t$ global maxima.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-03T15:17:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extremal configurations",
      "famous turan tetrahedron conjecture",
      "strong stability theorem",
      "free triple system",
      "first stability theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}