Byunghan Kim

In [9], Shelah introduced a class of first order theories, which he called simple, properly containing the class of stable theories. Here we prove for simple theories, (i) the equivalence of forking and dividing, (ii) the symmetry and transivity of forking.

Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T . canonical base of an amalgamation class is the union of names of ψ -definitions of , ψ ranging over stationary L -formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory hav

Abstract Let T be a countable, small simple theory. In this paper, we prove that for such T , the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

§1. Introduction . In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others. The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular algebraic structures which have

We discuss various conjectures and problems around the issue of when and whether stable formulas are responsible for forking in simple theories. We prove that if the simple theory $T$ has strong stable forking then any complete type is a nonforking extens

This paper presents an example of a supersimple nonlow theory and characterizes its independence relation.

This book is about simple first-order theories. The class of simple theories was introduced by S. Shelah in the early 1980s. Then several specific algebraic structures having simple theories have been studied by leading researchers, notably by E. Hrushovski. In the mid-1990s the author established in his thesis the symmetry and transitivity of non-forking for simple theories and, with A. Pillay, type-amalgamation for Lascar strong types. Since then a great deal of research work on simplicity the

The study of simple theories began with Shelah's paper "Simple unstable theories" where he introduced a class of first order theories, he called simple, having D(p,Δ, k) rank. The class includes all stable theories and some unstable theories. His intention was to ask whether we can build a theory of simple theories analogous to stability theory.

Abstract We study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p(x)=\operatorname {tp}(c/B)$ where $B=\operatorname {acl}(B)$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$ , the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure $\approx $ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove t

It is proved that the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan's theorem on a superstable theory.

Abstract We classify, in a nontrivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of 2-chains whose boundary is a 1-shell.