Itay Kaplan

The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density.

We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP_2.

In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss non-splintering, an interesting notion that appears in the work of Rami Grossberg, Andr\'es Villaveces and Monica VanDieren, and we show that it is not trivial (in the sense that it can be different than splitting) whenever the directionality of the theory is not small. In the appendix we study dense types in RCF.

We give a full solution to the question of existence of indiscernibles in dependent theories by proving the following theorem: for every $\theta$ there is a dependent theory $T$ of size $\theta$ such that for all $\kappa$ and $\delta$, $\kappa\to\left(\delta\right)_{T,1}$ iff $\kappa\to\left(\delta\right)_{\theta}^{<\omega}$. This means that unless there are good set theoretical reasons, there are large sets with no indiscernible sequences.

This paper can be viewed as a continuation of [KS09] that dealt with the automorphism tower problem without Choice. Here we deal with the inequation which connects the automorphism tower and the normalizer tower without Choice and introduce a new proof to a theorem of Fried and Koll\'ar that any group can be represented as an automorphism group of a field. The proof uses a simple construction: working more in graph theory, and less in algebra.

For a centerless group G, we can define its automorphism tower. We define G α : G 0 = G, G α+1 = Aut(G α ) and for limit ordinals \({G^{\delta}=\bigcup_{\alpha<\delta}G^{\alpha}}\) . Let τG be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says \({\tau_{G}<(2^{|G|})^{+}}\) and more. If we consider Thomas’ proof too set theoretical (using Fodor’s lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem without the Axiom of Choice. Moreover, we give a descriptive set theoretic approach for calculating an upper bound for τG for all countable groups G (better than the one an analysis of Thomas’ proof gives). We attach to every element in G α , the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from \({G*\langle x\rangle}\) ). This situation is generalized by defining “(G, A) is a special pair”.