Combinatorial background for non-structure

TL;DR

This paper compiles and discusses various combinatorial results related to trees, linear orders, pcf theory, and normal ideals, serving as a foundational reference for non-structure research.

Contribution

It provides a collection of relevant and sometimes new combinatorial results used in non-structure theory, including partition theorems and properties of linear orders and ideals.

Findings

Partition theorems on trees with omega levels

Properties of countable unions of scattered linear orders

Discussion of pcf theory and normal ideals

Abstract

This was supposed to be an appendix to the book "Non-structure", and probably will be if it materializes. It presents relevant material, sometimes new, which was used in works which were supposed to be part of that book. In section 1 we deal with partition theorems on trees with omega levels; it is self contained. In section 2 we deal with linear orders which are countable union of scattered ones with unary predicated, it is self contained. In section 3 we deal mainly with pcf theory but just quote. In section 4, on normal ideals, we repeat [Sh:247]. This is used in [Sh:331].