On ⊳ * -maximality

This paper investigates a connection between the ordering ⊳∗ among theories in model theory and the (N)SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1, and gives a connection between SOP1 and the maximality in the ⊳∗-ordering. Together with the known results about the connection between the (N)SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property. 1 1 This publication is numbered 692 in the list of publications of Saharon Shelah. The authors thank the United-States Israel Binational Science Foundation for their support