Forking and Dividing in Fields with Several Orderings and Valuations

Forking and dividing in Henson graphs

For $n\geq 3$, define $T_n$ to be the theory of the generic $K_n$-free graph, where $K_n$ is the complete graph on $n$ vertices. We prove a graph theoretic characterization of dividing in $T_n$, and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, $T_n$ provides a counterexample to a recent question of Chernikov and Kaplan.

阿贝尔群纯短正合列的扩张中分叉与分裂的转移原理

Transfer principles for forking and dividing in expansions of pure short exact sequences of Abelian groups

本研究探讨Aschenbrenner等人关于阿贝尔结构纯短正合列量词消去的辉煌Ax-Kochen-Ershov原理与分叉的关系，证明了该设定下分叉和分裂的Ax-Kochen-Ershov原理。

An independence theorem for NTP2 theories

We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski's terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 th…

A note on pseudofinite dimension and forking

In this paper we show that an instance of dividing in pseudofinite structures can be witnessed by a drop of the pseudofinite dimension. As an application of this result we give new proofs of known results for asymptotic classes of finite structures.

赋值域及相关结构中的分叉

Forking in valued fields and related structures

本论文研究赋值域及其约化结构中的分叉问题，为正则有序阿贝尔群建立几何描述，证明纯短正合序列扩张的Ax-Kochen-Ershov原理，并给出Henselian赋值域参数集成为扩张基的充分条件。

亨泽尔赋值域中的扩张基

Extension bases in Henselian valued fields

该研究针对剩余特征为零的亨泽尔赋值域，给出参数集成为扩张基的若干充分条件，突破了以往仅关注极大完全参数集的局限，还证明了p进数域超积的虚扩张中分叉与划分一致。