A class of graphs C is monadically stable if for any unary expansion Ĉ of C, one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from Ĉ. It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including graph classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs.
For several years, it has been suspected that one can create a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. During this talk I will give a characterization of monadic stability through the Flipper game: a combinatorial game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round localizes the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.ACM'17).
This is based on joint work with J. Gajarský, N. Mählmann, R. McCarty, P. Ohlmann, M. Pilipczuk, S. Siebertz, M. Sokołowski, and S. Toruńczyk.