Pseudo-finiteness of arbitrary graphs of bounded shrub-depth
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We consider classes of arbitrary (finite or infinite) graphs of bounded shrub-depth, specifically the classes TM_r(d) of arbitrary graphs that have tree models of height d and r labels. We show that the graphs of TM_r(d) are MSO-pseudo-finite relative to the class TM^f_r(d) of finite graphs of TM_r(d); that is, that every MSO sentence true in a graph of TM_r(d) is also true in a graph of TM^f_r(d). We also show that TM_r(d) is closed under ultraproducts and ultraroots. These results have two consequences. The first is that the index of the MSO[m]-equivalence relation on graphs of TM_r(d) is bounded by a (d+1)-fold exponential in m. The second is that TM_r(d) is exactly the class of all graphs that are MSO-pseudo-finite relative to TM^f_r(d).
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