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The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.
We don't know if the theorem 2 is true whenever p=ω , m=ω_1,n=ω_2 and n^p=ω_3=2^p : we do not suppose g.c.h. .