{
  "id": "1907.07639",
  "version": "v1",
  "published": "2019-07-17T16:53:55.000Z",
  "updated": "2019-07-17T16:53:55.000Z",
  "title": "A Tight Bound for Hyperaph Regularity",
  "authors": [
    "Guy Moshkovitz",
    "Asaf Shapira"
  ],
  "comment": "To appear in GAFA. This is a merged version of arXiv:1804.05511 and arXiv:1804.05513. See arXiv:1804.05511 for a self contained proof of the theorem for the special case of 3-uniform hypergraphs",
  "categories": [
    "math.CO"
  ],
  "abstract": "The hypergraph regularity lemma -- the extension of Szemer\\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-R\\\"odl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k \\ge 2$, thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers' famous lower bound for graph regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-17T16:53:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperaph regularity",
      "tight bound",
      "szemeredis graph regularity lemma",
      "yield regular partitions",
      "hypergraph regularity lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}