  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.
