---
_id: '15173'
abstract:
- lang: eng
  text: 'We show that the number of linear spaces on a set of n points and the number
    of rank-3 matroids on a ground set of size n are both of the form (cn+o(n))n2/6,
    where c=e3√/2−3(1+3–√)/2. This is the final piece of the puzzle for enumerating
    fixed-rank matroids at this level of accuracy: the numbers of rank-1 and rank-2
    matroids on a ground set of size n have exact representations in terms of well-known
    combinatorial functions, and it was recently proved by van der Hofstad, Pendavingh,
    and van der Pol that for constant r≥4 there are (e1−rn+o(n))nr−1/r! rank-r matroids
    on a ground set of size n. In our proof, we introduce a new approach for bounding
    the number of clique decompositions of a complete graph, using quasirandomness
    instead of the so-called entropy method that is common in this area.'
acknowledgement: "Sah and Sawhney were supported by NSF Graduate Research Fellowship
  Program DGE-1745302. Sah was supported by the PD Soros Fellowship.\r\nWe thank Michael
  Simkin for helpful comments on the manuscript. We thank Zach Hunter for\r\nseveral
  corrections."
article_processing_charge: Yes
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
citation:
  ama: Kwan MA, Sah A, Sawhney M. Enumerating matroids and linear spaces. <i>Comptes
    Rendus Mathematique</i>. 2023;361(G2):565-575. doi:<a href="https://doi.org/10.5802/crmath.423">10.5802/crmath.423</a>
  apa: Kwan, M. A., Sah, A., &#38; Sawhney, M. (2023). Enumerating matroids and linear
    spaces. <i>Comptes Rendus Mathematique</i>. Academie des Sciences. <a href="https://doi.org/10.5802/crmath.423">https://doi.org/10.5802/crmath.423</a>
  chicago: Kwan, Matthew Alan, Ashwin Sah, and Mehtaab Sawhney. “Enumerating Matroids
    and Linear Spaces.” <i>Comptes Rendus Mathematique</i>. Academie des Sciences,
    2023. <a href="https://doi.org/10.5802/crmath.423">https://doi.org/10.5802/crmath.423</a>.
  ieee: M. A. Kwan, A. Sah, and M. Sawhney, “Enumerating matroids and linear spaces,”
    <i>Comptes Rendus Mathematique</i>, vol. 361, no. G2. Academie des Sciences, pp.
    565–575, 2023.
  ista: Kwan MA, Sah A, Sawhney M. 2023. Enumerating matroids and linear spaces. Comptes
    Rendus Mathematique. 361(G2), 565–575.
  mla: Kwan, Matthew Alan, et al. “Enumerating Matroids and Linear Spaces.” <i>Comptes
    Rendus Mathematique</i>, vol. 361, no. G2, Academie des Sciences, 2023, pp. 565–75,
    doi:<a href="https://doi.org/10.5802/crmath.423">10.5802/crmath.423</a>.
  short: M.A. Kwan, A. Sah, M. Sawhney, Comptes Rendus Mathematique 361 (2023) 565–575.
date_created: 2024-03-24T23:01:00Z
date_published: 2023-02-01T00:00:00Z
date_updated: 2025-09-09T14:26:32Z
day: '01'
ddc:
- '510'
department:
- _id: MaKw
doi: 10.5802/crmath.423
external_id:
  arxiv:
  - '2112.03788'
  isi:
  - '001167671400009'
file:
- access_level: open_access
  checksum: d1d0e0a854a79ae95fb66d75d9117a68
  content_type: application/pdf
  creator: dernst
  date_created: 2024-03-25T07:21:52Z
  date_updated: 2024-03-25T07:21:52Z
  file_id: '15174'
  file_name: 2023_ComptesRendusMath_Kwan.pdf
  file_size: 598097
  relation: main_file
  success: 1
file_date_updated: 2024-03-25T07:21:52Z
fulldoi: https://doi.org/10.5802/crmath.423
has_accepted_license: '1'
intvolume: '       361'
isi: 1
issue: G2
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 565-575
publication: Comptes Rendus Mathematique
publication_identifier:
  eissn:
  - 1778-3569
  issn:
  - 1631-073X
publication_status: published
publisher: Academie des Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: Enumerating matroids and linear spaces
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 361
year: '2023'
...
---
OA_place: repository
OA_type: green
_id: '22182'
abstract:
- lang: eng
  text: We present a short new proof of the canonical polynomial van der Waerden theorem,
    recently established by Girão.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jacob
  full_name: Fox, Jacob
  last_name: Fox
- first_name: Yuval
  full_name: Wigderson, Yuval
  id: 2d0023a0-1567-11f0-833d-d5c1e476d4b5
  last_name: Wigderson
- first_name: Yufei
  full_name: Zhao, Yufei
  last_name: Zhao
citation:
  ama: Fox J, Wigderson Y, Zhao Y. A short proof of the canonical polynomial van der
    Waerden theorem. <i>Comptes Rendus Mathématique</i>. 2020;358(8):957-959. doi:<a
    href="https://doi.org/10.5802/crmath.101">10.5802/crmath.101</a>
  apa: Fox, J., Wigderson, Y., &#38; Zhao, Y. (2020). A short proof of the canonical
    polynomial van der Waerden theorem. <i>Comptes Rendus Mathématique</i>. Académie
    des Sciences. <a href="https://doi.org/10.5802/crmath.101">https://doi.org/10.5802/crmath.101</a>
  chicago: Fox, Jacob, Yuval Wigderson, and Yufei Zhao. “A Short Proof of the Canonical
    Polynomial van Der Waerden Theorem.” <i>Comptes Rendus Mathématique</i>. Académie
    des Sciences, 2020. <a href="https://doi.org/10.5802/crmath.101">https://doi.org/10.5802/crmath.101</a>.
  ieee: J. Fox, Y. Wigderson, and Y. Zhao, “A short proof of the canonical polynomial
    van der Waerden theorem,” <i>Comptes Rendus Mathématique</i>, vol. 358, no. 8.
    Académie des Sciences, pp. 957–959, 2020.
  ista: Fox J, Wigderson Y, Zhao Y. 2020. A short proof of the canonical polynomial
    van der Waerden theorem. Comptes Rendus Mathématique. 358(8), 957–959.
  mla: Fox, Jacob, et al. “A Short Proof of the Canonical Polynomial van Der Waerden
    Theorem.” <i>Comptes Rendus Mathématique</i>, vol. 358, no. 8, Académie des Sciences,
    2020, pp. 957–59, doi:<a href="https://doi.org/10.5802/crmath.101">10.5802/crmath.101</a>.
  short: J. Fox, Y. Wigderson, Y. Zhao, Comptes Rendus Mathématique 358 (2020) 957–959.
date_created: 2026-06-29T11:00:09Z
date_published: 2020-12-03T00:00:00Z
date_updated: 2026-07-14T09:19:20Z
day: '03'
doi: 10.5802/crmath.101
extern: '1'
external_id:
  arxiv:
  - '2005.04135'
fulldoi: https://doi.org/10.5802/crmath.101
intvolume: '       358'
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2005.04135
mathsc:
- 05D10
- 11B30
month: '12'
oa: 1
oa_version: Preprint
page: 957-959
publication: Comptes Rendus Mathématique
publication_identifier:
  eissn:
  - 1778-3569
  issn:
  - 1631-073X
publication_status: published
publisher: Académie des Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: A short proof of the canonical polynomial van der Waerden theorem
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 358
year: '2020'
...