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 to relational structures a result previously known for graphs 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 class of monotone structures that is not (monadically) NIP. This is a contribution towards the conjecture of Bonnet et al. that the hereditary classes of structures admitting fixed-parameter tractable model-checking are precisely those that are monadically NIP.

翻译：我们证明：对于任意有限关系结构的单调类，该类的初等理论在稳定性理论意义下是NIP的，当且仅当该类中结构的Gaifman图集合是处处稀疏的。这将对图论中已知结果推广至关系结构，并解答了Adler与Adler(2014)提出的未解决问题。该结论通过应用Ramsey理论技巧得以建立，揭示了NIP性质在单调类中具有高度稳健性。我们还证明：对于任何非（一元）NIP的单调结构类，其初等逻辑模型检验问题是难解的。这一结果对Bonnet等人提出的猜想——即具有固定参数易处理模型检验性质的可遗传结构类恰为一元NIP类——做出了贡献。