{
  "id": "2101.09769",
  "version": "v1",
  "published": "2021-01-24T18:47:25.000Z",
  "updated": "2021-01-24T18:47:25.000Z",
  "title": "A Removal Lemma for Ordered Hypergraphs",
  "authors": [
    "Henry Towsner"
  ],
  "categories": [
    "math.CO",
    "math.LO"
  ],
  "abstract": "We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph $(V,G,<)$ has few induced copies of a small ordered hypergraph $(W,H,\\prec)$ then there is a small modification $G'$ so that $(V,G',<)$ has no induced copies of $(W,H,\\prec)$. (Note that we do \\emph{not} need to modify the ordering $<$.) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of $\\sigma$-algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a ``very structured'' set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-24T18:47:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "induced hypergraph removal lemma",
      "simultaneously generalizing alon-ben-eliezer-fischers removal lemma",
      "induced copies",
      "ultraproduct limit object",
      "keisler graded probability space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}