Toward classifying unstable theories

The paper deals with two issues: the existence of universal models of a theory T and related properties when cardinal arithmetic does not give this existence offhand. In the first section we prove that simple theories (e.g., theories without the tree property, a class properly containing the stable theories) behaves ``better'' than theories with the strict order property, by criterion from [Sh:457]. In the second section we introduce properties SOP_n such that the strict order property implies SOP_{n+1}, which implies SOP_n, which in turn implies the tree property. Now SOP_4 already implies non-existence of universal models in cases where earlier the strict order property was needed, and SOP_3 implies maximality in the Keisler order, again improving an earlier result which had used the strict order property

Decomposition horizons: from graph sparsity to model-theoretic dividing lines

Let $\mathscr C$ be a hereditary class of graphs. Assume that for every $p$ there is a hereditary NIP class $\mathscr D_p$ with the property that the vertex set of every graph $G\in\mathscr C$ can be partitioned into $N_p=N_p(G)$ parts in such a way that the union of any $p$ parts induce a subgraph in $\mathscr D_p$ and $\log N_p(G)\in o(\log |G|)$. We prove that $\mathscr C$ is (monadically) NIP. Similarly, if every $\mathscr D_p$ is stable, then $\mathscr C$ is (monadically) stable. Results of this type lead to the definition of decomposition horizons as closure operators. We establish some of their basic properties and provide several further examples of decomposition horizons

Structural Properties of the First-Order Transduction Quasiorder

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of additive classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most k, for k ? 1 form a strict hierarchy in the FO transduction quasiorder