Borel Order Dimension

TL;DR

This paper introduces the concept of Borel order dimension for Borel quasi orders, establishing dichotomies and structural properties, and explores its relation to Turing degrees and set-theoretic models.

Contribution

It defines Borel order dimension, proves a generalized dichotomy for Borel dichromatic number, and analyzes the structure of Borel quasi orders, especially in relation to Turing degrees.

Findings

Borel order dimension is closely related to Borel dichromatic number.

A dichotomy generalizing the ${ ext{G}_0}$-dichotomy is established for Borel simple directed graphs.

Borel quasi orders of countable Borel dimension are Borel linearizable.

Abstract

We introduce and study a notion of Borel order dimension for Borel quasi orders. It will be shown that this notion is closely related to the notion of Borel dichromatic number for simple directed graphs. We prove a dichotomy, which generalizes the ${\GGG}_{0}$-dichotomy, for the Borel dichromatic number of Borel simple directed graphs. By applying this dichotomy to Borel quasi orders, another dichotomy that characterizes the Borel quasi orders of uncountable Borel dimension is proved. We obtain further structural information about the Borel quasi orders of countable Borel dimension by showing that they are all Borel linearizable. We then investigate the locally countable Borel quasi orders in more detail, paying special attention to the Turing degrees, and produce models of set theory where the continuum is arbitrarily large and all locally countable Borel quasi orders are of Borel dimension less than the continuum. Combining our results here with earlier work shows that the Borel order dimension of the Turing degrees is usually strictly larger than its classical order dimension.