Document Open Access Logo

A Structure Theorem for Pseudo-Segments and Its Applications

Authors Jacob Fox, János Pach, Andrew Suk

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.59.pdf
  • Filesize: 0.69 MB
  • 14 pages

Document Identifiers

Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, CA, USA
János Pach
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

Funding

  • Fox, Jacob: Supported by NSF Award DMS-2154129.
  • Pach, János: Rényi Institute, Budapest. Supported by NKFIH grant K-176529, Austrian Science Fund Z 342-N31 and ERC Advanced Grant "GeoScape."
  • Suk, Andrew: Supported by NSF CAREER award DMS-1800746, and NSF awards DMS-1952786 and DMS-2246847.

Cite As Get BibTex

Jacob Fox, János Pach, and Andrew Suk. A Structure Theorem for Pseudo-Segments and Its Applications. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.59

Abstract

We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such that almost all of the bipartite graphs between pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most n(log n)^O(log k) edges.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Regularity lemma
  • pseudo-segments
  • intersection graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger. Twisted ways to find plane structures in simple drawings of complete graphs. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 5:1-5:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  2. Noga Alon, János Pach, Rom Pinchasi, Rados Radoicic, and Micha Sharir. Crossing patterns of semi-algebraic sets. J. Comb. Theory, Ser. A, 111(2):310-326, 2005. Google Scholar
  3. Grant Cairns and Yury Nikolayevsky. Bounds for generalized thrackles. Discrete Comput. Geom., 23(2):191-206, 2000. Google Scholar
  4. Jacob Fox and János Pach. Separator theorems and Turán-type results for planar intersection graphs. Adv. Math., 219:1070-1080, 2008. Google Scholar
  5. Jacob Fox and János Pach. A separator theorem for string graphs and its applications. Comb. Probab. Comput., 19(3):371-390, 2010. Google Scholar
  6. Jacob Fox, János Pach, and Andrew Suk. A polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing. SIAM J. Comput., 45(6):2199-2223, 2016. Google Scholar
  7. Jacob Fox, János Pach, and Andrew Suk. Enumeration of intersection graphs of x-monotone curves. In preparation, 2024+. Google Scholar
  8. Jacob Fox, János Pach, and Andrew Suk. A structure theorem for pseudo-segments and its applications. arxiv, 2024. URL: https://arxiv.org/abs/2312.01028.
  9. Jacob Fox, János Pach, and Csaba D. Tóth. Intersection patterns of curves. J. Lond. Math. Soc., 83(2):389-406, 2011. Google Scholar
  10. Jacob Fox and Benny Sudakov. Density theorems for bipartite graphs and related Ramsey-type results. Combinatorica, 29(2):153-196, 2009. Google Scholar
  11. Radoslav Fulek and János Pach. A computational approach to Conway’s thrackle conjecture. Comput. Geom., 44(6-7):345-355, 2011. Google Scholar
  12. Radoslav Fulek and János Pach. Thrackles: An improved upper bound. Discrete Appl. Math., 259:226-231, 2019. Google Scholar
  13. Jacob E. Goodman. Proof of a conjecture of Burr, Grünbaum, and Sloane. Discrete Math., 32(1):27-35, 1980. Google Scholar
  14. Jacob E. Goodman and Richard Pollack. Semispaces of configurations, cell complexes of arrangements. J. Comb. Theory, Ser. A, 37(3):257-293, 1984. Google Scholar
  15. Sariel Har-Peled. Constructing planar cuttings in theory and practice. SIAM J. Comput., 29(6):2016-2039, 2000. Google Scholar
  16. János Komlós, Ali Shokoufandeh, Miklós Simonovits, and Endre Szemerédi. The regularity lemma and its applications in graph theory. In Gholamreza B. Khosrovshahi, Ali Shokoufandeh, and Mohammad Amin Shokrollahi, editors, Theoretical Aspects of Computer Science, Advanced Lectures (First Summer School on Theoretical Aspects of Computer Science, Tehran, Iran, July 2000), volume 2292 of Lecture Notes in Computer Science, pages 84-112. Springer, 2000. Google Scholar
  17. László Lovász, János Pach, and Mario Szegedy. On Conway’s thrackle conjecture. Discrete Comput. Geom., 18(4):369-376, 1997. Google Scholar
  18. János Pach and József Solymosi. Structure theorems for systems of segments. In Jin Akiyama, Mikio Kano, and Masatsugu Urabe, editors, Discrete and Computational Geometry, Japanese Conference, JCDCG 2000, Tokyo, Japan, November, 22-25, 2000, Revised Papers, volume 2098 of Lecture Notes in Computer Science, pages 308-317. Springer, 2000. Google Scholar
  19. János Pach and József Solymosi. Crossing patterns of segments. J. Comb. Theory, Ser. A, 96(2):316-325, 2001. Google Scholar
  20. János Pach and Géza Tóth. Disjoint edges in topological graphs. In Jin Akiyama, Edy Tri Baskoro, and Mikio Kano, editors, Combinatorial Geometry and Graph Theory, Indonesia-Japan Joint Conference,IJCCGGT 2003, Bandung, Indonesia, September 13-16, 2003, Revised Selected Papers, volume 3330 of Lecture Notes in Computer Science, pages 133-140. Springer, 2003. Google Scholar
  21. Lisa Sauermann. On the speed of algebraically defined graph classes. Adv. Math., 380:107593, 2021. Google Scholar
  22. Andrew Suk. Density theorems for intersection graphs of t-monotone curves. SIAM J. Discrete Math., 27(3):1323-1334, 2013. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact