Contributions to neostability theory
File(s)
Author(s)
Miguel Gómez, Alberto
Type
Thesis
Abstract
This thesis makes several contributions to the study of structure theory for NSOP1 and NSOP4 theories, as well as the theories of abstract independence relations and patterns in neostability theory. Firstly, we provide a partial answer to a question asked independently by d'Elbée and Kim and show that, under the assumption of the stable Kim-forking conjecture, every NSOP1 rosy theory must be simple. We also prove that the theory of a Frobenius field has stable Kim-forking.
Secondly, we provide two proofs, one combinatorial and another using abstract independence relations, of the model-theoretic classification of the countable homogeneous H4-free 3-hypertournament studied by Cherlin, Hubička, Konečný, and Nešetřil. Our main result regarding this structure is that its theory is SOP3, TP2, and NSOP4.
Thirdly, to contextualise the phenomena observed previously more broadly, we develop a framework, in the style of Adler, for studying the notion of "witnessing" (appearing as variants of Kim's Lemma in the literature) as a binary relation between abstract independence relations. This involves the relativisation of the notions of Kim- and Conant-independence. Our notion of witnessing allows us to obtain several results from simplicity, NSOP1, NTP2, etc., as instances of general theorems in this abstract framework.
We finish this thesis by systematically exploring the notions of coding and realising graphs appearing in the work of Bailetti, Laskowski, and Shelah, and provide new characterisations of stability, NIP, NSOP3, NPM(2), NIPn, and NFOPn using these tools. Alongside this, we further develop the theory of positive patterns in connection with realisability, and introduce the notion of duality of patterns to formally capture the relationship between syntactic properties such as SOP2 and ATP.
Secondly, we provide two proofs, one combinatorial and another using abstract independence relations, of the model-theoretic classification of the countable homogeneous H4-free 3-hypertournament studied by Cherlin, Hubička, Konečný, and Nešetřil. Our main result regarding this structure is that its theory is SOP3, TP2, and NSOP4.
Thirdly, to contextualise the phenomena observed previously more broadly, we develop a framework, in the style of Adler, for studying the notion of "witnessing" (appearing as variants of Kim's Lemma in the literature) as a binary relation between abstract independence relations. This involves the relativisation of the notions of Kim- and Conant-independence. Our notion of witnessing allows us to obtain several results from simplicity, NSOP1, NTP2, etc., as instances of general theorems in this abstract framework.
We finish this thesis by systematically exploring the notions of coding and realising graphs appearing in the work of Bailetti, Laskowski, and Shelah, and provide new characterisations of stability, NIP, NSOP3, NPM(2), NIPn, and NFOPn using these tools. Alongside this, we further develop the theory of positive patterns in connection with realisability, and introduce the notion of duality of patterns to formally capture the relationship between syntactic properties such as SOP2 and ATP.
Version
Open Access
Date Issued
2026-03-19
Date Awarded
2026-06-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Evans, David
Kestner, Charlotte
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/W524323/1
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)