We generalise the concept of clique width to structures of arbitrary signature and cardinality. We present characterisations of clique width in terms of decompositions of a structure and via interpretations in trees. Several model-theoretic properties of clique width are investigated including VC-dimension and preservation of finite clique width under elementary extensions and compactness.

Review by: Achim Blumensath The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 394-396, September 2013.

We study tree-like decompositions of models of a theory and a related complexity measure called partition width. We prove a dichotomy concerning partition width and definable pairing functions: either the partition width of models is bounded, or the theory admits definable pairing functions. Our proof rests on structure results concerning indiscernible sequences and finitely satisfiable types for theories without definable pairing functions. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Aiming for applications in monadic second-order model theory, we study first-order theories without definable pairing functions. Our main results concern forking-properties of sequences of indiscernibles. These turn out to be very well-behaved for the theories under consideration.