### More on ${\rm SOP}_1$ and ${\rm SOP}_2$

by Shelah and Usvyatsov. [ShUs:844]

Annals Pure and Applied Logic, 2008

This paper continues [DjSh692]. We present a rank
function for NSOP_{1} theories and give an example of a
theory which is NSOP_{1} but not simple. We also
investigate the connection between maximality in the
ordering lhd^* among complete first order theories
and the (N)SOP {}_2 property. We complete the proof
started in
[DjSh692] of the fact that
lhd^*-maximality implies SOP {}_2 and get weaker
results in the other direction. The paper provides a step
toward the classification of unstable theories without the
strict order property.

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