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Abstract We initiate a systematic study ofgeneric stability independenceand introduce the class oftreeless theoriesin which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP$$_{3}$$theory is simple. 

We experimentally observe signatures of 475-fs-long sech-squared-shaped solitons in a ps-pumped phase-mismatched parametric oscillator in the normal dispersion regime, purely due to cascaded quadratic nonlinearities. The results are in good agreement with our theoretical predictions. 

We experimentally observe the formation of dissipative cavity solitons at the boundaries of a topological lattice. Our work reveals new opportunities to study both nonlinear topological photonics and dissipative cavity solitons in coupled resonator arrays. 

An interpretation is an operation that maps an input graph to an output graph by redefining its edge relation using a first-order formula. This rich framework includes operations such as taking the complement or a fixed power of a graph as (very) special cases. We prove that there is an FPT algorithm for the first-order model checking problem on classes of graphs which are first-order interpretable in classes of graphs with bounded local cliquewidth. Notably, this includes interpretations of planar graphs, and of classes of bounded genus in general. To obtain this result we develop a new tool which works in a very general setting of NIP classes and which we believe can be an important ingredient in obtaining similar results in the future. 

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, Bollobás, 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.