{
  "abstract": "Fix an integer $r\\ge 2$ and a finite simple $r$-uniform hypergraph $F$ with at least one edge and no isolated vertices. An $n$-vertex $r$-graph is $ε$-far from being $F$-free if at least $εn^r$ edges must be deleted to destroy every copy of $F$. A finite $r$-graph $H$ is $F$-abundant if there are constants $c,C>0$ such that every sufficiently large $ε$-far host contains at least $cε^C n^{v(H)}$ labelled copies of $H$. A family is $F$-abundant when one member has this lower bound in each host, although the member may depend on the host and on $ε$, while $c$ and $C$ are common to the family. We prove that every abundant family contains an abundant member. We prove the analogous coloured theorem for $F$-partite hosts containing edge-disjoint part-respecting copies of $F$ such that every vertex lies in at least $εn^{r-1}$ of them. The case $r=2$ yields the coloured and uncoloured graph compactness theorems, answers Question 5.2 of Girão, Hurley, Illingworth and Michel, and proves their Conjecture 5.1 [J. Lond. Math. Soc., 2024]. We also obtain an explicit bound $\\lfloor 2r(C+1)\\rfloor$ for the order of the non-isolated core of a selected witness. Moreover, we give several applications. For example, we construct translation-invariant linear systems from abundant coloured hypergraphs, obtain a square-root bound for an equation associated with a cycle of bounded length, give a one-sided tester based on one fixed graph when distance from the property gives a polynomial lower bound on distance from being $F$-free, and prove that no algorithm decides whether a family of finite simple graphs enumerated by a Turing machine is $K_3$-abundant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.21640",
  "author_total": 4,
  "authors": [
    {
      "name": "Shuang Sun"
    },
    {
      "name": "Yan Wang"
    },
    {
      "name": "Yuyao Yang"
    },
    {
      "name": "Jiasheng Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2b0pem0y0cmn2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.21640",
  "fulltext_url": null,
  "id": "2b0pem0y0cmn2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.21640",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7171409693",
  "pdf_url": "https://arxiv.org/pdf/2607.21640",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2b0pem0y0cmn2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.21640",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.21640"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Compactness of abundance in asymmetric hypergraph removal lemmas",
  "title_text": "Compactness of abundance in asymmetric hypergraph removal lemmas",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2b0pem0y0cmn2",
  "venue": "arXiv",
  "versions": []
}