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Monadic NIP in Monotone Classes of Relational Structures Braunfeld, Samuel Dawar, Anuj Eleftheriadis, Ioannis Papadopoulos, Aris Model theory finite model theory structural graph theory model-checking 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Samuel Braunfeld and Anuj Dawar and Ioannis Eleftheriadis and Aris Papadopoulos 2023 Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2023.119 urn:nbn:de:0030-drops-181712 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.119 eng https://creativecommons.org/licenses/by/4.0/legalcode