Interpreting nowhere dense graph classes as a classical notion of model theory

Hans Adler, Isolde Adler
2014 European journal of combinatorics (Print)  
A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of complete graphs that occur as r-minors. We observe that this recent tameness notion from (algorithmic) graph theory is essentially the earlier stability theoretic notion of superflatness. For subgraph-closed classes of graphs we prove equivalence to stability and to not having the independence property. Expressed in terms of PAC learning, the concept classes definable in first-order logic in a
more » ... subgraph-closed graph class have bounded sample complexity, if and only if the class is nowhere dense.
doi:10.1016/j.ejc.2013.06.048 fatcat:cmgyymeggvdxfkkhh5lhy5ugaa