More on SOP₁ and SOP₂

This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 (1996) 229-255] and [M. Džamonja, S. Shelah, On {white left-pointing small triangle}^*-maximality, Annals of Pure and Applied Logic 125 (2004) 119-158]. We present a rank function for NSOP₁ theories and give an example of a theory which is NSOP₁ but not simple. We also investigate the connection between maximality in the ordering {white left-pointing small triangle}^* among complete first order theories and the (N)SOP₂ property. We prove that {white left-pointing small triangle}^*-maximality implies SOP₂ and obtain certain results in the other direction. The paper provides a step toward the classification of unstable theories without the strict order property.