Strict Superstablity and Decidability of Certain Generic Graphs

TL;DR

This work builds ultraflat Hrushovski-Fraïssé generic structures from acyclic graphs to realize strictly superstable theories with tunable $U$-rank. Using predimension $δ(A)=|A|-|R[A]|$ and a full amalgamation framework over classes $\mathcal{K}_{α}$, it constructs Fraïssé limits $\\mathfrak{M}_{α}$ and a universal theory $UNIV_{α}$, then analyzes definability via closure formulas. Quantifier elimination down to closure formulas is established through closure types $cltp^{M}(\bar a)$ and a robust back-and-forth system, enabling precise control of forking and independence. The main results show each $Th(\\mathfrak{M}_{α})$ is strictly superstable, with specific $U$-rank and 1-based behavior depending on $α$, and the theories are decidable and pseudofinite, highlighting a tunable stability ladder within graph-based Hrushovski constructions.

Abstract

We show that the Hrushovski-\fraisse limit of certain classes of trees lead to strictly superstable theories of various U-ranks. In fact, for each $ α\inω+1\backslash\{0\} $ we introduce a strictly superstable theory of U-rank $ α. $ Furthermore, we show that these theories are decidable and pseudofinite.

Theorems & Definitions (51)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Corollary 2.7