A conjecture in algorithmic model theory predicts that the model-checking problem for first-order logic is fixed-parameter tractable on a hereditary graph class if and only if the class is monadically dependent. Originating in model theory, this notion is defined in terms of logic, and encompasses nowhere dense classes, monadically stable classes, and classes of bounded twin-width. Working towards this conjecture, we provide the first two combinatorial characterizations of monadically dependent graph classes. This yields the following dichotomy. On the structure side, we characterize monadic dependence by a Ramsey-theoretic property called flip-breakability. This notion generalizes the notions of uniform quasi-wideness, flip-flatness, and bounded grid rank, which characterize nowhere denseness, monadic stability, and bounded twin-width, respectively, and played a key role in their respective model checking algorithms. Natural restrictions of flip-breakability additionally characterize bounded treewidth and cliquewidth and bounded treedepth and shrubdepth. On the non-structure side, we characterize monadic dependence by explicitly listing few families of forbidden induced subgraphs. This result is analogous to the characterization of nowhere denseness via forbidden subdivided cliques, and allows us to resolve one half of the motivating conjecture: First-order model checking is AW[$*$]-hard on every hereditary graph class that is monadically independent. The result moreover implies that hereditary graph classes which are small, have almost bounded twin-width, or have almost bounded flip-width, are monadically dependent. Lastly, we lift our result to also obtain a combinatorial dichotomy in the more general setting of monadically dependent classes of binary structures.

翻译：算法模型理论中的一个猜想预测：对于一阶逻辑的模型检测问题，在遗传图类上具有固定参数可跟踪性当且仅当该图类是一元依赖的。这一概念源于模型论，通过逻辑定义，涵盖无处稠密类、一元稳定类以及有界双宽度类。针对这一猜想，我们首次给出了两个关于一元依赖图类的组合刻画，从而得到以下二分性结论。在结构层面，我们通过一种称为"可翻转有界性"的拉姆齐理论性质来刻画一元依赖性。该概念统一了刻画无处稠密性、一元稳定性和有界双宽度的均匀拟宽性、可翻转平坦性和有界网格秩等性质，这些性质在各自的模型检测算法中均发挥关键作用。可翻转有界性的自然限制形式还可分别刻画有界树宽与团宽、有界树深与灌木深。在非结构层面，我们通过明确列举少数几个禁止诱导子图族来刻画一元依赖性。这一结果类似于通过禁止细分团来刻画无处稠密性，并使我们得以解决该猜想的一半：在每一个一元独立的遗传图类上，一阶逻辑模型检测是AW[$*$]-困难的。该结果还表明，具有小规模、几乎有界双宽度或几乎有界可翻转宽度的遗传图类必然是一元依赖的。最后，我们将结果推广至更一般的二元结构一元依赖类，同样获得了组合二分性。