Twin-Width IV: Ordered Graphs and Matrices*

We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory. This has several consequences. First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh, Bollobas, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Finally, it settles our small conjecture [SODA '21] in the case of ordered graphs.