Extremal combinatorics in discrete structues

• Project/Area Number: 18K03399
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: University of the Ryukyus
• Principal Investigator: Tokushige Norihide 琉球大学, 教育学部, 教授 (00217481)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 組合せ論 / 離散極値構造 / 交差族 / ハイパーグラフ / スペクトラルグラフ理論 / 極値離散構造 / ランダムグラフ / 線形計画問題 / スライスランク法

Outline of Final Research Achievements

In this research project, we studied the extremal structures appeared in extremal combinatorics, and also discussed methods for this purpose. The two main results are as follows. We obtained an upper bound for the size of a set without a given "shape" in a vector space over a finite fields. We determined the extremal structure of the largest multiply intersecting families by using the corresponding spectral information.

Academic Significance and Societal Importance of the Research Achievements

上に述べた成果のうち、(1)については「非退化な」解を含まない集合に関する研究において、より強い「非退化性」を定式化し解析したことに意義がある。この視点はこの分野に新しい方向を与え、その後、SauermannやEllenbergによってさらに発展している。交差族を調べる手法はいろいろあるが、現在のところ(2)の成果は固有値を用いる手法によってのみ得られており、この点に特徴がある。

Research Products

• [Int'l Joint Research] Kyungpook National University/Duksung Women’s University(韓国)
• [Int'l Joint Research] Kyungpook National University/Duksung Women’s University(韓国)
• [Journal Article] Non-trivial 3-wise intersecting uniform families (2023)
  • Author(s): Tokushige Norihide
  • Journal Title: Discrete Mathematics Volume: 346 Issue: 5 Pages: 113368-113368
  • DOI: 10.1016/j.disc.2023.113368
  • Peer Reviewed
• [Journal Article] Application of hypergraph Hoffman’s bound to intersecting families (2022)
  • Author(s): Tokushige Norihide
  • Journal Title: Algebraic Combinatorics Volume: 5 Issue: 3 Pages: 537-557
  • DOI: 10.5802/alco.222
  • Peer Reviewed / Open Access
• [Journal Article] Extending Muirhead’s Inequality (2021)
  • Author(s): Kato Mitsuo、Kosuda Masashi、Tokushige Norihide
  • Journal Title: Graphs and Combinatorics Volume: 37 Issue: 5 Pages: 1923-1941
  • DOI: 10.1007/s00373-021-02356-z
  • Peer Reviewed
• [Journal Article] Solving linear equations in a vector space over a finite field (2021)
  • Author(s): Mimura Masato、Tokushige Norihide
  • Journal Title: Discrete Mathematics Volume: 344 Issue: 12 Pages: 112603-112603
  • DOI: 10.1016/j.disc.2021.112603
  • Peer Reviewed
• [Journal Article] When are stars the largest cross-intersecting families? (2020)
  • Author(s): Tokushige, Norihide
  • Journal Title: Discrete Math. Volume: 343 Issue: 2 Pages: 111645-111645
  • DOI: 10.1016/j.disc.2019.111645
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] AK-type stability theorems on cross t-intersecting families (2019)
  • Author(s): Lee, Sang June; Siggers, Mark; Tokushige, Norihide
  • Journal Title: European J. Combin. Volume: 82 Pages: 102993-102993
  • DOI: 10.1016/j.ejc.2019.07.004
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] 大きな有限に現れる構造をめぐって (2019)
  • Author(s): 徳重典英
  • Journal Title: 現代思想 Volume: 47
• [Presentation] 円と直線のなす配置 (2023)
  • Author(s): 徳重典英
  • Organizer: 研究会「直観幾何学2023」
• [Presentation] 方程式の解に関する組合せ論の紹介 (2022)
  • Author(s): 徳重典英
  • Organizer: 2022早稲田整数論研究集会
  • Invited
• [Presentation] Solving linear equations in a vector space (2020)
  • Author(s): Norihide Tokushige
  • Organizer: スペクトラルグラフ理論および周辺領域第9回研究集会
• [Presentation] Some applications of hypergraph Hoffman's bound (2020)
  • Author(s): Norihide Tokushige
  • Organizer: Extremal set systems, zoom webinar at Renyi Institute
  • Int'l Joint Research / Invited
• [Presentation] Recent applications of the slice rank method (2019)
  • Author(s): Norihide Tokushige
  • Organizer: RIMS共同研究「代数的組合せ論と関連する群と代数の研究」
  • Invited
• [Presentation] Muirhead's inequality and a maximal flow (2019)
  • Author(s): Norihide Tokushige
  • Organizer: The 92nd KPPY Combinatorics Seminar (Daegu)
  • Int'l Joint Research / Invited
• [Presentation] Stability results in intersecting families (2018)
  • Author(s): Norihide Tokushige
  • Organizer: The 20th KIAS Combinatorics workshop
  • Int'l Joint Research / Invited
• [Book] Extremal problems for finite sets (2018)
  • Author(s): Peter Frankl, Norihide Tokushige
  • Publisher: American Mathematical Society
  • ISBN: 9781470440398