Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity

@misc{li2026strong,
  title = {{Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity}},
  author = {Heng Li and Hong Liu and Xizhi Liu and Jing Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.26735},
  eprint = {2606.26735},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.26735},
}

TY  - UNPB
TI  - Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity
AU  - Heng Li
AU  - Hong Liu
AU  - Xizhi Liu
AU  - Jing Wang
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2606.26735
UR  - https://arxiv.org/abs/2606.26735
AB  - The supersaturation problem asks, for a fixed r-graph ℱ, for the minimum number of copies of ℱ in an n-vertex r-graph with ex(n,ℱ)+q edges. Mubayi conjectured a local form of supersaturation under a stability hypothesis: if ℱ is non-r-partite and stable, meaning roughly that the extremal ℱ-free construction is unique and all near-extremal ℱ-free r-graphs are close to it, then this minimum should be at least q c(n,ℱ), where c(n,ℱ) is the minimum number of copies created by adding one edge to the extremal ℱ-free r-graph. We disprove this conjectured local lower bound in the strongest possible quantitative sense that the ratio to the conjectured lower bound can be made arbitrarily small. More precisely, for every r≥2 and every K>1, we construct a stable non-r-partite r-graph ℱ such that, for all sufficiently large n and every 1≤ q≤δn, there is an n-vertex r-graph with ex(n,ℱ)+q edges containing at most K^−1q c(n,ℱ) copies of ℱ. This arbitrarily large failure already occurs when q=1.
ER  -