General non-structure theory and constructing from linear orders

TL;DR

This paper develops a general framework for constructing complex structures from linear orders and other index models, demonstrating how certain properties lead to many non-isomorphic models, with specific results for linear orders.

Contribution

It introduces a non-structure theory framework for building complex models from index models like linear orders and trees, and shows how certain embedding and omission properties produce many non-isomorphic models.

Findings

Constructs many non-isomorphic models from linear orders with specific properties.

Establishes a framework for building structures from index models with complicatedness properties.

Provides applications demonstrating the theory's effectiveness.

Abstract

The theme of the first two sections, is to prepare the framework of how from a ``complicated'' family of so called index models $I \in K_1$ we build many and/or complicated structures in a class $K_2$. The index models are characteristically linear orders, trees with $\kappa+1$ levels (possibly with linear order on the set of successors of a member) and linearly ordered graphs; for this we formulate relevant complicatedness properties (called bigness). In the third section we show stronger results concerning linear orders. If for each linear order $I$ of cardinality $\lambda > \aleph_0$ we can attach a model $M_I \in K_\lambda$ in which the linear order can be embedded such that for enough cuts of $I$, their being omitted is reflected in $M_I$, then there are $2^\lambda$ non-isomorphic cases. We also do the work for some applications.