{
  "id": "2308.09382",
  "version": "v1",
  "published": "2023-08-18T08:25:09.000Z",
  "updated": "2023-08-18T08:25:09.000Z",
  "title": "Hypergraphs with irrational Turán density and many extremal configurations",
  "authors": [
    "Jianfeng Hou",
    "Heng Li",
    "Guanghui Wang",
    "Yixiao Zhang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Unlike graphs, determining Tur\\'{a}n densities of hypergraphs is known to be notoriously hard in general. The essential reason is that for many classical families of $r$-uniform hypergraphs $\\mathcal{F}$, there are perhaps many near-extremal $\\mathcal{M}_t$-free configurations with very different structure. Such a phenomenon is called not stable, and Liu and Mubayi gave a first not stable example. Another perhaps reason is that little is known about the set consisting of all possible Tur\\'{a}n densities which has cardinality of the continuum. Let $t\\ge 2$ be an integer. In this paper, we construct a finite family $\\mathcal{M}_t$ of 3-uniform hypergraphs such that the Tur\\'{a}n density of $\\mathcal{M}_t$ is irrational, and there are $t$ near-extremal $\\mathcal{M}_t$-free configurations that are far from each other in edit-distance. This is the first not stable example that has an irrational Tur\\'{a}n density. It also provides a new phenomenon about feasible region functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-18T08:25:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irrational turán density",
      "extremal configurations",
      "free configurations",
      "stable example",
      "near-extremal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}