Age and weak indivisibility

Countable “Urysohn metric spaces” (see the next section), with finite distance sets, are indivisible (see Sauer (2013)  [19]). Countable Urysohn metric spaces are age indivisible, which follows from the Hales-Jewett theorem (Hales and Jewett (1963)  [7]; see Delhommé et al. (2007)  [2]). For non-Urysohn homogeneous countable metric spaces, except for the example mentioned above, the weak indivisibility question is completely open. We will show, in this paper, that a countable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible.