Sparsity and dimension

摘要: We prove that posets of bounded height whose cover graphs belong to a fixed class with expansion have dimension. Bounded expansion, introduced by Ne\v{s}et\v{r}il and Ossona de Mendez as model for sparsity in graphs, is property naturally satisfied wide range graph classes, from structure theory (graphs excluding minor or topological minor) drawing (e.g. book thickness). Therefore, our theorem generalizes number results including the most recent one minor. also show result sense best possible, it does not extend nowhere dense classes; fact, already fails locally treewidth.