# Ordering metric spaces and Assouad Nagata dimension

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Anna Erschler, École Normale Supérieure
Fine Hall 314

We show that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true (joint with I. Mitrofanov). One of such conditions can be described in terms of a spectral gap for a sequence of graphs, and another one is the property of containing weakly cubes of arbitrary large dimension. We discuss then spaces which admits extremely good ordering (as good as for metric trees).