# Coherent forests

## TL;DR

This paper explores the properties of coherent forests, a generalization of trees, demonstrating their construction within ZFC and methods to obtain Suslin forests through forcing and combinatorial principles.

## Contribution

It introduces the concept of coherence in forests, shows how to construct coherent Aronszajn forests in ZFC, and develops new forcing techniques to obtain coherent Suslin forests.

## Key findings

- Coherent Aronszajn forests can be constructed within ZFC.
- Forcing methods can produce coherent Suslin forests.
- A combinatorial principle similar to diamond can generate Suslin forests from large cardinals.

## Abstract

A forest is a generalization of a tree, and here we consider the Aronszajn and Suslin properties for forests. We focus on those forests satisfying coherence, a local smallness property. We show that coherent Aronszajn forests can be constructed within ZFC. We give several ways of obtaining coherent Suslin forests by forcing, one of which generalizes the well-known argument of Todor\v{c}evi\'{c} that a Cohen real adds a Suslin tree. Another uses a strong combinatorial principle that plays a similar role to diamond. We show that, starting from a large cardinal, this principle can be obtained by a forcing that is small relative to the forest it constructs.