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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