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Monadic NIP in Monotone Classes of Relational Structures

Authors Samuel Braunfeld , Anuj Dawar , Ioannis Eleftheriadis , Aris Papadopoulos

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Author Details

Samuel Braunfeld
  • Computer Science Institute of Charles University (IUUK), Prague, Czech Republic
Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Ioannis Eleftheriadis
  • Department of Computer Science and Technology, University of Cambridge, UK
Aris Papadopoulos
  • School of Mathematics, Univesity of Leeds, UK


We want to thank the referees for their numerous improvements to the text, and for suggesting the simplified argument in Section 6.

Cite AsGet BibTex

Samuel Braunfeld, Anuj Dawar, Ioannis Eleftheriadis, and Aris Papadopoulos. Monadic NIP in Monotone Classes of Relational Structures. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 119:1-119:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We prove that for any monotone class of finite relational structures, the first-order theory of the class is NIP in the sense of stability theory if, and only if, the collection of Gaifman graphs of structures in this class is nowhere dense. This generalises results previously known for graphs to relational structures and answers an open question posed by Adler and Adler (2014). The result is established by the application of Ramsey-theoretic techniques and shows that the property of being NIP is highly robust for monotone classes. We also show that the model-checking problem for first-order logic is intractable on any monotone class of structures that is not (monadically) NIP. This is a contribution towards the conjecture that the hereditary classes of structures admitting fixed-parameter tractable model-checking are precisely those that are monadically NIP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Mathematics of computing → Combinatorics
  • Model theory
  • finite model theory
  • structural graph theory
  • model-checking


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  1. Hans Adler and Isolde Adler. Interpreting nowhere dense graph classes as a classical notion of model theory. European Journal of Combinatorics, 36:322-330, 2014. URL:
  2. John T. Baldwin. Fundamentals of Stability Theory. Perspectives in Logic. Cambridge University Press, 2017. Google Scholar
  3. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, Stéphan Thomassé, and Szymon Toruńczyk. Twin-width IV: Ordered graphs and matrices, 2021. URL:
  4. Samuel Braunfeld and Michael C. Laskowski. Existential characterizations of monadic NIP, 2022. URL:
  5. Ashok K. Chandra and Philip M. Merlin. Optimal implementation of conjunctive queries in relational data bases. In John E. Hopcroft, Emily P. Friedman, and Michael A. Harrison, editors, Proceedings of the 9th Annual ACM Symposium on Theory of Computing, May 4-6, 1977, Boulder, Colorado, USA, pages 77-90. ACM, 1977. URL:
  6. Anuj Dawar. Finite model theory on tame classes of structures. In MFCS, volume 4708 of Lecture Notes in Computer Science, pages 2-12. Springer, 2007. Google Scholar
  7. Paul Erdős and Richard Rado. A combinatorial theorem. Journal of the London Mathematical Society, 1(4):249-255, 1950. Google Scholar
  8. Jakub Gajarský, Michal Pilipczuk, and Szymon Torunczyk. Stable graphs of bounded twin-width. In Christel Baier and Dana Fisman, editors, LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Haifa, Israel, August 2 - 5, 2022, pages 39:1-39:12. ACM, 2022. URL:
  9. Jakub Gajarský, Petr Hliněný, Daniel Lokshtanov, Jan Obdržálek, and M. S. Ramanujan. A new perspective on FO model checking of dense graph classes, 2018. URL:
  10. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. J. ACM, 64(3), June 2017. URL:
  11. Wilfrid Hodges. Model Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1993. URL:
  12. Alexandr Kostochka, Dhruv Mubayi, and Jacques Verstraëte. Turán problems and shadows II: Trees. Journal of Combinatorial Theory, Series B, 122:457-478, 2017. URL:
  13. Stephan Kreutzer and Anuj Dawar. Parameterized complexity of first-order logic. Electron. Colloquium Comput. Complex., TR09-131, 2009. URL:
  14. Jaroslav Nešetřil and Patrice Ossona De Mendez. On nowhere dense graphs. European Journal of Combinatorics, 32(4):600-617, 2011. Google Scholar
  15. Jaroslav Nešetřil and Patrice Ossona De Mendez. Sparsity: graphs, structures, and algorithms, volume 28. Springer Science & Business Media, 2012. Google Scholar
  16. Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. Parameterized circuit complexity of model-checking on sparse structures. In Anuj Dawar and Erich Grädel, editors, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 789-798. ACM, 2018. URL:
  17. Klaus-Peter Podewski and Martin Ziegler. Stable graphs. Fund. Math, 100(2):101-107, 1978. Google Scholar
  18. F. P. Ramsey. On a Problem of Formal Logic. Proceedings of the London Mathematical Society, s2-30(1):264-286, January 1930. URL:
  19. Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New York, 1978. Google Scholar
  20. Pierre Simon. A Guide to NIP Theories. Lecture Notes in Logic. Cambridge University Press, 2015. URL:
  21. Pierre Simon and Szymon Toruńczyk. Ordered graphs of bounded twin-width, 2021. URL:
  22. Algorithms, logic, and structure workshop in Warwick, open problem session. URL: Accessed: 2023-05-02.
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