---
OA_place: repository
OA_type: green
_id: '22163'
abstract:
- lang: eng
  text: "For a field F and integers d and k, a set A ⊆ Fd is called k-nearly orthogonal
    if its\r\nmembers are non-self-orthogonal and every k + 1 vectors of A include
    an orthogonal pair.\r\nWe prove that for every prime p there exists some δ = δ(p)>
    0, such that for every field\r\nF of characteristic p and for all integers k ≥
    2 and d ≥ k, there exists a k-nearly orthogonal\r\nset of at least dδ·k/ logk
    vectors of Fd. The size of the set is optimal up to the logk term\r\nin the exponent.
    We further prove two extensions of this result. In the first, we provide a\r\nlarge
    set A of non-self-orthogonal vectors of Fd such that for every two subsets of
    A of\r\nsize k+1 each, some vector of one of the subsets is orthogonal to some
    vector of the other.\r\nIn the second extension, every k + 1 vectors of the produced
    set A include ℓ + 1 pairwise\r\northogonal vectors for an arbitrary fixed integer
    1 ≤ ℓ ≤ k. The proofs involve probabilistic\r\nand spectral arguments and the
    hypergraph container method"
article_number: '114373'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Ishay
  full_name: Haviv, Ishay
  last_name: Haviv
- first_name: Sam
  full_name: Mattheus, Sam
  last_name: Mattheus
- first_name: Aleksa
  full_name: Milojević, Aleksa
  last_name: Milojević
- first_name: Yuval
  full_name: Wigderson, Yuval
  id: 2d0023a0-1567-11f0-833d-d5c1e476d4b5
  last_name: Wigderson
citation:
  ama: Haviv I, Mattheus S, Milojević A, Wigderson Y. Larger nearly orthogonal sets
    over finite fields. <i>Discrete Mathematics</i>. 2025;348(4). doi:<a href="https://doi.org/10.1016/j.disc.2024.114373">10.1016/j.disc.2024.114373</a>
  apa: Haviv, I., Mattheus, S., Milojević, A., &#38; Wigderson, Y. (2025). Larger
    nearly orthogonal sets over finite fields. <i>Discrete Mathematics</i>. Elsevier.
    <a href="https://doi.org/10.1016/j.disc.2024.114373">https://doi.org/10.1016/j.disc.2024.114373</a>
  chicago: Haviv, Ishay, Sam Mattheus, Aleksa Milojević, and Yuval Wigderson. “Larger
    Nearly Orthogonal Sets over Finite Fields.” <i>Discrete Mathematics</i>. Elsevier,
    2025. <a href="https://doi.org/10.1016/j.disc.2024.114373">https://doi.org/10.1016/j.disc.2024.114373</a>.
  ieee: I. Haviv, S. Mattheus, A. Milojević, and Y. Wigderson, “Larger nearly orthogonal
    sets over finite fields,” <i>Discrete Mathematics</i>, vol. 348, no. 4. Elsevier,
    2025.
  ista: Haviv I, Mattheus S, Milojević A, Wigderson Y. 2025. Larger nearly orthogonal
    sets over finite fields. Discrete Mathematics. 348(4), 114373.
  mla: Haviv, Ishay, et al. “Larger Nearly Orthogonal Sets over Finite Fields.” <i>Discrete
    Mathematics</i>, vol. 348, no. 4, 114373, Elsevier, 2025, doi:<a href="https://doi.org/10.1016/j.disc.2024.114373">10.1016/j.disc.2024.114373</a>.
  short: I. Haviv, S. Mattheus, A. Milojević, Y. Wigderson, Discrete Mathematics 348
    (2025).
das_tickbox: '1'
date_created: 2026-06-29T10:53:05Z
date_published: 2025-04-01T00:00:00Z
date_updated: 2026-07-14T08:17:17Z
day: '01'
doi: 10.1016/j.disc.2024.114373
extern: '1'
external_id:
  arxiv:
  - '2404.01057'
intvolume: '       348'
issue: '4'
keyword:
- Nearly orthogonal sets
- Ramsey theory
- Finite fields
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: 'https://doi.org/10.48550/arXiv.2404.01057 '
month: '04'
oa: 1
oa_version: Preprint
publication: Discrete Mathematics
publication_identifier:
  issn:
  - 0012-365X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Larger nearly orthogonal sets over finite fields
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 348
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '22165'
abstract:
- lang: eng
  text: "The book graph \U0001D435(\U0001D458)\r\n\U0001D45B consists of \U0001D45B
    copies of \U0001D43E\U0001D458+1 joined along a common \U0001D43E\U0001D458. In
    the prequel to this paper, we studied the diagonal Ramsey number \U0001D45F⁡(\U0001D435(\U0001D458)\r\n\U0001D45B,\U0001D435(\U0001D458)\r\n\U0001D45B).
    Here we consider the natural off-diagonal variant \U0001D45F⁡(\U0001D435(\U0001D458)\r\n\U0001D450⁢\U0001D45B,\U0001D435(\U0001D458)\r\n\U0001D45B)
    for fixed \U0001D450 ∈(0,1]. In this more general setting, we show that an interesting
    dichotomy emerges: for very small \U0001D450, a simple \U0001D458-partite construction
    dictates the Ramsey function and all nearly-extremal colourings are close to being
    \U0001D458-partite, while, for \U0001D450 bounded away from 0, random colourings
    of an appropriate density are asymptotically optimal and all nearly-extremal colourings
    are quasirandom. Our investigations also open up a range of questions about what
    happens for intermediate values of \U0001D450.\r\n\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: David
  full_name: Conlon, David
  last_name: Conlon
- 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
citation:
  ama: Conlon D, Fox J, Wigderson Y. Off-diagonal book Ramsey numbers. <i>Combinatorics,
    Probability and Computing</i>. 2023;32(3):516-545. doi:<a href="https://doi.org/10.1017/s0963548322000360">10.1017/s0963548322000360</a>
  apa: Conlon, D., Fox, J., &#38; Wigderson, Y. (2023). Off-diagonal book Ramsey numbers.
    <i>Combinatorics, Probability and Computing</i>. Cambridge University Press. <a
    href="https://doi.org/10.1017/s0963548322000360">https://doi.org/10.1017/s0963548322000360</a>
  chicago: Conlon, David, Jacob Fox, and Yuval Wigderson. “Off-Diagonal Book Ramsey
    Numbers.” <i>Combinatorics, Probability and Computing</i>. Cambridge University
    Press, 2023. <a href="https://doi.org/10.1017/s0963548322000360">https://doi.org/10.1017/s0963548322000360</a>.
  ieee: D. Conlon, J. Fox, and Y. Wigderson, “Off-diagonal book Ramsey numbers,” <i>Combinatorics,
    Probability and Computing</i>, vol. 32, no. 3. Cambridge University Press, pp.
    516–545, 2023.
  ista: Conlon D, Fox J, Wigderson Y. 2023. Off-diagonal book Ramsey numbers. Combinatorics,
    Probability and Computing. 32(3), 516–545.
  mla: Conlon, David, et al. “Off-Diagonal Book Ramsey Numbers.” <i>Combinatorics,
    Probability and Computing</i>, vol. 32, no. 3, Cambridge University Press, 2023,
    pp. 516–45, doi:<a href="https://doi.org/10.1017/s0963548322000360">10.1017/s0963548322000360</a>.
  short: D. Conlon, J. Fox, Y. Wigderson, Combinatorics, Probability and Computing
    32 (2023) 516–545.
date_created: 2026-06-29T10:53:47Z
date_published: 2023-05-01T00:00:00Z
date_updated: 2026-07-14T08:27:40Z
day: '01'
doi: 10.1017/s0963548322000360
extern: '1'
external_id:
  arxiv:
  - '2110.14483'
intvolume: '        32'
issue: '3'
keyword:
- Ramsey theory
- book graphs
- Ramsey goodness
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2110.14483
mathsc:
- 05C55
- 05D10
month: '05'
oa: 1
oa_version: Preprint
page: 516-545
publication: Combinatorics, Probability and Computing
publication_identifier:
  eissn:
  - 1469-2163
  issn:
  - 0963-5483
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Off-diagonal book Ramsey numbers
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2023'
...