Compactness of abundance in asymmetric hypergraph removal lemmas

Abstract

Fix an integer r≥ 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 ⌊ 2r(C+1)⌋ 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 K₃-abundant.

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BibTeX RIS

BibTeX

@misc{sun2026compactness,
  title = {{Compactness of abundance in asymmetric hypergraph removal lemmas}},
  author = {Shuang Sun and Yan Wang and Yuyao Yang and Jiasheng Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.21640},
  eprint = {2607.21640},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.21640},
}

RIS

TY  - UNPB
TI  - Compactness of abundance in asymmetric hypergraph removal lemmas
AU  - Shuang Sun
AU  - Yan Wang
AU  - Yuyao Yang
AU  - Jiasheng Zeng
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2607.21640
UR  - https://arxiv.org/abs/2607.21640
AB  - Fix an integer r≥ 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 ⌊ 2r(C+1)⌋ 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.
ER  -

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