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PreprintCC BY 4.0arXiv

An Improved Upper Bound for Recovering Pairs

  • Simone Costa
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Abstract

We prove that every recovering pair (π’œ,ℬ) of nonempty families of subsets of an n-element set satisfies |π’œ||ℬ|≀(223949/100000)n=(2.23949)n. A result of Mond, Souza and Versteegen, combined with the sharp (9/4)n bound of Fang and Huang for cancellative pairs, gives (2.2499)n for recovering pairs. We improve this to (2.23949)n. The proof combines two entropy inequalities. One is derived directly from the recovering property, while the other applies the Fang-Huang bound to families obtained from the original pair by a construction of Mond, Souza and Versteegen. The two inequalities are then combined by a convex combination.

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Cite

BibTeXRIS
BibTeX
@misc{costa2026improved,
  title = {{An Improved Upper Bound for Recovering Pairs}},
  author = {Simone Costa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03851},
  eprint = {2610.03851},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03851},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An Improved Upper Bound for Recovering Pairs
AU  - Simone Costa
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03851
UR  - https://arxiv.org/abs/2610.03851
AB  - We prove that every recovering pair (π’œ,ℬ) of nonempty families of subsets of an n-element set satisfies |π’œ||ℬ|≀(223949/100000)^n=(2.23949)^n. A result of Mond, Souza and Versteegen, combined with the sharp (9/4)^n bound of Fang and Huang for cancellative pairs, gives (2.2499)^n for recovering pairs. We improve this to (2.23949)^n. The proof combines two entropy inequalities. One is derived directly from the recovering property, while the other applies the Fang-Huang bound to families obtained from the original pair by a construction of Mond, Souza and Versteegen. The two inequalities are then combined by a convex combination.
ER  -