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In recent years, much work has been done to measure and compare thecomplexity of orbit equivalence relations, especially for certain classes ofPolish groups. We start by introducing some language to organize this previouswork, namely the notion of classification strength of Polish groups. Broadlyspeaking, a Polish group $G$ has stronger classification strength than $H$ ifevery orbit equivalence relation induced by a continuous action of $H$ on aPolish space can be "emulated" by such an action of $G$ in the sense of Borelreduction.  Among the non-Archimedean Polish groups, the groups with the highestclassification strength are those that involve $S_\infty$, the Polish group ofpermutations of a countably-infinite set. We prove that several properties,including a weakening of the disjoint amalgamation in Fra\"{i}ss\'{e} theory, aweakening of the existence of an absolute set of generating indiscernibles, andnot having ordinal rank for a particular coanalytic rank function, are allequivalent to a non-Archimedean Polish group involving $S_\infty$. Furthermore,we show the equivalence relation $=^+$, which is a relatively simple benchmarkequivalence relation in the theory of Borel reducibility, can only beclassified by such groups that involve $S_\infty$.

Classification Strength of Polish Groups I: Involving $S_\infty$