---
_id: '12680'
abstract:
- lang: eng
  text: The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting
    family of r-element subsets of  was extended to the setting of exterior algebra
    in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not
    been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado
    theorem and the characterization of the equality case therein, as well as those
    of the Hilton–Milner theorem to the setting of exterior algebra in the simplest
    non-trivial case of two-forms follow from a folklore puzzle about possible arrangements
    of an intersecting family of lines.
article_number: '113363'
article_processing_charge: No
article_type: letter_note
arxiv: 1
author:
- first_name: Grigory
  full_name: Ivanov, Grigory
  id: 87744F66-5C6F-11EA-AFE0-D16B3DDC885E
  last_name: Ivanov
- first_name: Seyda
  full_name: Köse, Seyda
  id: 8ba3170d-dc85-11ea-9058-c4251c96a6eb
  last_name: Köse
citation:
  ama: Ivanov G, Köse S. Erdős-Ko-Rado and Hilton-Milner theorems for two-forms. <i>Discrete
    Mathematics</i>. 2023;346(6). doi:<a href="https://doi.org/10.1016/j.disc.2023.113363">10.1016/j.disc.2023.113363</a>
  apa: Ivanov, G., &#38; Köse, S. (2023). Erdős-Ko-Rado and Hilton-Milner theorems
    for two-forms. <i>Discrete Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/j.disc.2023.113363">https://doi.org/10.1016/j.disc.2023.113363</a>
  chicago: Ivanov, Grigory, and Seyda Köse. “Erdős-Ko-Rado and Hilton-Milner Theorems
    for Two-Forms.” <i>Discrete Mathematics</i>. Elsevier, 2023. <a href="https://doi.org/10.1016/j.disc.2023.113363">https://doi.org/10.1016/j.disc.2023.113363</a>.
  ieee: G. Ivanov and S. Köse, “Erdős-Ko-Rado and Hilton-Milner theorems for two-forms,”
    <i>Discrete Mathematics</i>, vol. 346, no. 6. Elsevier, 2023.
  ista: Ivanov G, Köse S. 2023. Erdős-Ko-Rado and Hilton-Milner theorems for two-forms.
    Discrete Mathematics. 346(6), 113363.
  mla: Ivanov, Grigory, and Seyda Köse. “Erdős-Ko-Rado and Hilton-Milner Theorems
    for Two-Forms.” <i>Discrete Mathematics</i>, vol. 346, no. 6, 113363, Elsevier,
    2023, doi:<a href="https://doi.org/10.1016/j.disc.2023.113363">10.1016/j.disc.2023.113363</a>.
  short: G. Ivanov, S. Köse, Discrete Mathematics 346 (2023).
date_created: 2023-02-26T23:01:00Z
date_published: 2023-06-01T00:00:00Z
date_updated: 2023-10-04T11:54:57Z
day: '01'
department:
- _id: UlWa
- _id: GradSch
doi: 10.1016/j.disc.2023.113363
external_id:
  arxiv:
  - '2201.10892'
intvolume: '       346'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2201.10892'
month: '06'
oa: 1
oa_version: Preprint
publication: Discrete Mathematics
publication_identifier:
  issn:
  - 0012-365X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
  record:
  - id: '13331'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Erdős-Ko-Rado and Hilton-Milner theorems for two-forms
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 346
year: '2023'
...
---
_id: '6638'
abstract:
- lang: eng
  text: The crossing number of a graph G is the least number of crossings over all
    possible drawings of G. We present a structural characterization of graphs with
    crossing number one.
article_processing_charge: No
arxiv: 1
author:
- first_name: 'André '
  full_name: 'Silva, André '
  last_name: Silva
- first_name: Alan M
  full_name: Arroyo Guevara, Alan M
  id: 3207FDC6-F248-11E8-B48F-1D18A9856A87
  last_name: Arroyo Guevara
  orcid: 0000-0003-2401-8670
- first_name: Bruce
  full_name: Richter, Bruce
  last_name: Richter
- first_name: Orlando
  full_name: Lee, Orlando
  last_name: Lee
citation:
  ama: Silva A, Arroyo Guevara AM, Richter B, Lee O. Graphs with at most one crossing.
    <i>Discrete Mathematics</i>. 2019;342(11):3201-3207. doi:<a href="https://doi.org/10.1016/j.disc.2019.06.031">10.1016/j.disc.2019.06.031</a>
  apa: Silva, A., Arroyo Guevara, A. M., Richter, B., &#38; Lee, O. (2019). Graphs
    with at most one crossing. <i>Discrete Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/j.disc.2019.06.031">https://doi.org/10.1016/j.disc.2019.06.031</a>
  chicago: Silva, André , Alan M Arroyo Guevara, Bruce Richter, and Orlando Lee. “Graphs
    with at Most One Crossing.” <i>Discrete Mathematics</i>. Elsevier, 2019. <a href="https://doi.org/10.1016/j.disc.2019.06.031">https://doi.org/10.1016/j.disc.2019.06.031</a>.
  ieee: A. Silva, A. M. Arroyo Guevara, B. Richter, and O. Lee, “Graphs with at most
    one crossing,” <i>Discrete Mathematics</i>, vol. 342, no. 11. Elsevier, pp. 3201–3207,
    2019.
  ista: Silva A, Arroyo Guevara AM, Richter B, Lee O. 2019. Graphs with at most one
    crossing. Discrete Mathematics. 342(11), 3201–3207.
  mla: Silva, André, et al. “Graphs with at Most One Crossing.” <i>Discrete Mathematics</i>,
    vol. 342, no. 11, Elsevier, 2019, pp. 3201–07, doi:<a href="https://doi.org/10.1016/j.disc.2019.06.031">10.1016/j.disc.2019.06.031</a>.
  short: A. Silva, A.M. Arroyo Guevara, B. Richter, O. Lee, Discrete Mathematics 342
    (2019) 3201–3207.
date_created: 2019-07-14T21:59:20Z
date_published: 2019-11-01T00:00:00Z
date_updated: 2023-08-29T06:31:41Z
day: '01'
department:
- _id: UlWa
doi: 10.1016/j.disc.2019.06.031
ec_funded: 1
external_id:
  arxiv:
  - '1901.09955'
  isi:
  - '000486358100025'
intvolume: '       342'
isi: 1
issue: '11'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1901.09955
month: '11'
oa: 1
oa_version: Preprint
page: 3201-3207
project:
- _id: 26366136-B435-11E9-9278-68D0E5697425
  name: Reglas de Conectividad funcional en el hipocampo
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: Discrete Mathematics
publication_identifier:
  issn:
  - 0012-365X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Graphs with at most one crossing
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 342
year: '2019'
...
---
_id: '4065'
abstract:
- lang: eng
  text: We prove that given n⩾3 convex, compact, and pairwise disjoint sets in the
    plane, they may be covered with n non-overlapping convex polygons with a total
    of not more than 6n−9 sides, and with not more than 3n−6 distinct slopes. Furthermore,
    we construct sets that require 6n−9 sides and 3n−6 slopes for n⩾3. The upper bound
    on the number of slopes implies a new bound on a recently studied transversal
    problem.
acknowledgement: 'The first author acknowledges the support by Amoco Fnd. Fat. Dev.
  Comput. Sci. l-6-44862. Work on this paper by the second author was supported by
  a Shell Fellowship in Computer Science. The third author as supported by the office
  of Naval Research under grant NOOO14-86K-0416. '
article_processing_charge: No
article_type: original
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Arch
  full_name: Robison, Arch
  last_name: Robison
- first_name: Xiao
  full_name: Shen, Xiao
  last_name: Shen
citation:
  ama: Edelsbrunner H, Robison A, Shen X. Covering convex sets with non-overlapping
    polygons. <i>Discrete Mathematics</i>. 1990;81(2):153-164. doi:<a href="https://doi.org/10.1016/0012-365X(90)90147-A">10.1016/0012-365X(90)90147-A</a>
  apa: Edelsbrunner, H., Robison, A., &#38; Shen, X. (1990). Covering convex sets
    with non-overlapping polygons. <i>Discrete Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/0012-365X(90)90147-A">https://doi.org/10.1016/0012-365X(90)90147-A</a>
  chicago: Edelsbrunner, Herbert, Arch Robison, and Xiao Shen. “Covering Convex Sets
    with Non-Overlapping Polygons.” <i>Discrete Mathematics</i>. Elsevier, 1990. <a
    href="https://doi.org/10.1016/0012-365X(90)90147-A">https://doi.org/10.1016/0012-365X(90)90147-A</a>.
  ieee: H. Edelsbrunner, A. Robison, and X. Shen, “Covering convex sets with non-overlapping
    polygons,” <i>Discrete Mathematics</i>, vol. 81, no. 2. Elsevier, pp. 153–164,
    1990.
  ista: Edelsbrunner H, Robison A, Shen X. 1990. Covering convex sets with non-overlapping
    polygons. Discrete Mathematics. 81(2), 153–164.
  mla: Edelsbrunner, Herbert, et al. “Covering Convex Sets with Non-Overlapping Polygons.”
    <i>Discrete Mathematics</i>, vol. 81, no. 2, Elsevier, 1990, pp. 153–64, doi:<a
    href="https://doi.org/10.1016/0012-365X(90)90147-A">10.1016/0012-365X(90)90147-A</a>.
  short: H. Edelsbrunner, A. Robison, X. Shen, Discrete Mathematics 81 (1990) 153–164.
date_created: 2018-12-11T12:06:44Z
date_published: 1990-04-15T00:00:00Z
date_updated: 2022-02-22T15:45:55Z
day: '15'
doi: 10.1016/0012-365X(90)90147-A
extern: '1'
intvolume: '        81'
issue: '2'
language:
- iso: eng
main_file_link:
- url: https://www.sciencedirect.com/science/article/pii/0012365X9090147A?via%3Dihub
month: '04'
oa_version: None
page: 153 - 164
publication: Discrete Mathematics
publication_identifier:
  eissn:
  - 1872-681X
  issn:
  - 0012-365X
publication_status: published
publisher: Elsevier
publist_id: '2060'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Covering convex sets with non-overlapping polygons
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 81
year: '1990'
...
---
_id: '4107'
abstract:
- lang: eng
  text: A set of m planes dissects E3 into cells, facets, edges and vertices. Letting
    deg(c) be the number of facets that bound a cellc, we give exact and asymptotic
    bounds on the maximum of ∈cinCdeg(c), if C is a family of cells of the arrangement
    with fixed cardinality.
acknowledgement: 'Research reported in the paper was conducted while the second author
  was visiting the Technical University of Graz. Support provided by the Technical
  University for this visit is gratefully acknowledged. '
article_processing_charge: No
article_type: original
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: David
  full_name: Haussler, David
  last_name: Haussler
citation:
  ama: Edelsbrunner H, Haussler D. The complexity of cells in 3-dimensional arrangements.
    <i>Discrete Mathematics</i>. 1986;60(C):139-146. doi:<a href="https://doi.org/10.1016/0012-365X(86)90008-7">10.1016/0012-365X(86)90008-7</a>
  apa: Edelsbrunner, H., &#38; Haussler, D. (1986). The complexity of cells in 3-dimensional
    arrangements. <i>Discrete Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/0012-365X(86)90008-7">https://doi.org/10.1016/0012-365X(86)90008-7</a>
  chicago: Edelsbrunner, Herbert, and David Haussler. “The Complexity of Cells in
    3-Dimensional Arrangements.” <i>Discrete Mathematics</i>. Elsevier, 1986. <a href="https://doi.org/10.1016/0012-365X(86)90008-7">https://doi.org/10.1016/0012-365X(86)90008-7</a>.
  ieee: H. Edelsbrunner and D. Haussler, “The complexity of cells in 3-dimensional
    arrangements,” <i>Discrete Mathematics</i>, vol. 60, no. C. Elsevier, pp. 139–146,
    1986.
  ista: Edelsbrunner H, Haussler D. 1986. The complexity of cells in 3-dimensional
    arrangements. Discrete Mathematics. 60(C), 139–146.
  mla: Edelsbrunner, Herbert, and David Haussler. “The Complexity of Cells in 3-Dimensional
    Arrangements.” <i>Discrete Mathematics</i>, vol. 60, no. C, Elsevier, 1986, pp.
    139–46, doi:<a href="https://doi.org/10.1016/0012-365X(86)90008-7">10.1016/0012-365X(86)90008-7</a>.
  short: H. Edelsbrunner, D. Haussler, Discrete Mathematics 60 (1986) 139–146.
date_created: 2018-12-11T12:06:59Z
date_published: 1986-06-01T00:00:00Z
date_updated: 2022-02-01T12:44:50Z
day: '01'
doi: 10.1016/0012-365X(86)90008-7
extern: '1'
intvolume: '        60'
issue: C
language:
- iso: eng
month: '06'
oa_version: None
page: 139 - 146
publication: Discrete Mathematics
publication_identifier:
  eissn:
  - 1872-681X
  issn:
  - 0012-365X
publication_status: published
publisher: Elsevier
publist_id: '2019'
quality_controlled: '1'
status: public
title: The complexity of cells in 3-dimensional arrangements
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 60
year: '1986'
...