---
OA_place: repository
OA_type: green
_id: '18559'
abstract:
- lang: eng
  text: We prove a 1973 conjecture due to Erdős on the existence of Steiner triple
    systems with arbitrarily high girth.
acknowledgement: Sah and Sawhney were supported by NSF Graduate Research Fellowship
  Program DGE1745302. Sah was supported by the PD Soros Fellowship. Simkin was supported
  by the Center of Mathematical Sciences and Applications at Harvard University.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Matthew Alan
  full_name: Kwan, Matthew Alan
  id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
  last_name: Kwan
  orcid: 0000-0002-4003-7567
- first_name: Ashwin
  full_name: Sah, Ashwin
  last_name: Sah
- first_name: Mehtaab
  full_name: Sawhney, Mehtaab
  last_name: Sawhney
- first_name: Michael
  full_name: Simkin, Michael
  last_name: Simkin
citation:
  ama: Kwan MA, Sah A, Sawhney M, Simkin M. High-girth Steiner triple systems. <i>Annals
    of Mathematics</i>. 2024;200(3):1059-1156. doi:<a href="https://doi.org/10.4007/annals.2024.200.3.4">10.4007/annals.2024.200.3.4</a>
  apa: Kwan, M. A., Sah, A., Sawhney, M., &#38; Simkin, M. (2024). High-girth Steiner
    triple systems. <i>Annals of Mathematics</i>. Princeton University. <a href="https://doi.org/10.4007/annals.2024.200.3.4">https://doi.org/10.4007/annals.2024.200.3.4</a>
  chicago: Kwan, Matthew Alan, Ashwin Sah, Mehtaab Sawhney, and Michael Simkin. “High-Girth
    Steiner Triple Systems.” <i>Annals of Mathematics</i>. Princeton University, 2024.
    <a href="https://doi.org/10.4007/annals.2024.200.3.4">https://doi.org/10.4007/annals.2024.200.3.4</a>.
  ieee: M. A. Kwan, A. Sah, M. Sawhney, and M. Simkin, “High-girth Steiner triple
    systems,” <i>Annals of Mathematics</i>, vol. 200, no. 3. Princeton University,
    pp. 1059–1156, 2024.
  ista: Kwan MA, Sah A, Sawhney M, Simkin M. 2024. High-girth Steiner triple systems.
    Annals of Mathematics. 200(3), 1059–1156.
  mla: Kwan, Matthew Alan, et al. “High-Girth Steiner Triple Systems.” <i>Annals of
    Mathematics</i>, vol. 200, no. 3, Princeton University, 2024, pp. 1059–156, doi:<a
    href="https://doi.org/10.4007/annals.2024.200.3.4">10.4007/annals.2024.200.3.4</a>.
  short: M.A. Kwan, A. Sah, M. Sawhney, M. Simkin, Annals of Mathematics 200 (2024)
    1059–1156.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-11-17T23:01:48Z
date_published: 2024-11-01T00:00:00Z
date_updated: 2026-10-16T10:12:22Z
day: '01'
department:
- _id: MaKw
doi: 10.4007/annals.2024.200.3.4
external_id:
  arxiv:
  - '2201.04554'
  isi:
  - '001366233800004'
fulldoi: https://doi.org/10.4007/annals.2024.200.3.4
intvolume: '       200'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2201.04554
month: '11'
oa: 1
oa_version: Preprint
page: 1059-1156
publication: Annals of Mathematics
publication_identifier:
  eissn:
  - 1939-8980
  issn:
  - 0003-486X
publication_status: published
publisher: Princeton University
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: yes
title: High-girth Steiner triple systems
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 200
year: '2024'
...
---
OA_place: publisher
OA_type: free access
_id: '2357'
abstract:
- lang: eng
  text: 'The classic Poincaré inequality bounds the L q-norm of a function f in a
    bounded domain Ω ⊂ ℝ n in terms of some L p-norm of its gradient in Ω. We generalize
    this in two ways: In the first generalization we remove a set Τ from Ω and concentrate
    our attention on Λ = Ω \ Τ. This new domain might not even be connected and hence
    no Poincaré inequality can generally hold for it, or if it does hold it might
    have a very bad constant. This is so even if the volume of Τ is arbitrarily small.
    A Poincaré inequality does hold, however, if one makes the additional assumption
    that f has a finite L p gradient norm on the whole of Ω, not just on Λ. The important
    point is that the Poincaré inequality thus obtained bounds the L q-norm of f in
    terms of the L p gradient norm on Λ (not Ω) plus an additional term that goes
    to zero as the volume of Τ goes to zero. This error term depends on Τ only through
    its volume. Apart from this additive error term, the constant in the inequality
    remains that of the ''nice'' domain Ω. In the second generalization we are given
    a vector field A and replace ∇ by ∇ + iA(x) (geometrically, a connection on a
    U(1) bundle). Unlike the A = 0 case, the infimum of ∥(∇ + iA)f∥ p over all f with
    a given ∥f∥ q is in general not zero. This permits an improvement of the inequality
    by the addition of a term whose sharp value we derive. We describe some open problems
    that arise from these generalizations.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Élliott
  full_name: Lieb, Élliott
  last_name: Lieb
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
- first_name: Jakob
  full_name: Yngvason, Jakob
  last_name: Yngvason
citation:
  ama: Lieb É, Seiringer R, Yngvason J. Poincaré inequalities in punctured domains.
    <i>Annals of Mathematics</i>. 2003;158(3):1067-1080. doi:<a href="https://doi.org/10.4007/annals.2003.158.1067
    ">10.4007/annals.2003.158.1067 </a>
  apa: Lieb, É., Seiringer, R., &#38; Yngvason, J. (2003). Poincaré inequalities in
    punctured domains. <i>Annals of Mathematics</i>. Princeton University Press. <a
    href="https://doi.org/10.4007/annals.2003.158.1067 ">https://doi.org/10.4007/annals.2003.158.1067
    </a>
  chicago: Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “Poincaré Inequalities
    in Punctured Domains.” <i>Annals of Mathematics</i>. Princeton University Press,
    2003. <a href="https://doi.org/10.4007/annals.2003.158.1067 ">https://doi.org/10.4007/annals.2003.158.1067
    </a>.
  ieee: É. Lieb, R. Seiringer, and J. Yngvason, “Poincaré inequalities in punctured
    domains,” <i>Annals of Mathematics</i>, vol. 158, no. 3. Princeton University
    Press, pp. 1067–1080, 2003.
  ista: Lieb É, Seiringer R, Yngvason J. 2003. Poincaré inequalities in punctured
    domains. Annals of Mathematics. 158(3), 1067–1080.
  mla: Lieb, Élliott, et al. “Poincaré Inequalities in Punctured Domains.” <i>Annals
    of Mathematics</i>, vol. 158, no. 3, Princeton University Press, 2003, pp. 1067–80,
    doi:<a href="https://doi.org/10.4007/annals.2003.158.1067 ">10.4007/annals.2003.158.1067
    </a>.
  short: É. Lieb, R. Seiringer, J. Yngvason, Annals of Mathematics 158 (2003) 1067–1080.
date_created: 2018-12-11T11:57:11Z
date_published: 2003-11-01T00:00:00Z
date_updated: 2026-05-28T08:25:42Z
day: '01'
doi: '10.4007/annals.2003.158.1067 '
extern: '1'
external_id:
  arxiv:
  - '0205088'
fulldoi: 'https://doi.org/10.4007/annals.2003.158.1067 '
intvolume: '       158'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://arxiv.org/abs/math/0205088
month: '11'
oa: 1
oa_version: Published Version
page: 1067 - 1080
publication: Annals of Mathematics
publication_identifier:
  eissn:
  - 1939-8980
  issn:
  - 0003-486X
publication_status: published
publisher: Princeton University Press
publist_id: '4570'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Poincaré inequalities in punctured domains
type: journal_article
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 158
year: '2003'
...