Borel chain conditions of Borel posets

TL;DR

This paper investigates the hierarchy of chain conditions in Borel partial orders within descriptive combinatorics, revealing a distinct structure unlike the classical Borel case.

Contribution

It establishes a new hierarchy of chain conditions for Borel posets, highlighting differences from classical Borel combinatorics.

Findings

Identifies a hierarchy of chain conditions in Borel posets

Shows differences between Borel and classical chain conditions

Provides a framework for classifying Borel partial orders

Abstract

We study the coarse classification of partial orderings using chain conditions in the context of descriptive combinatorics. We show that (unlike the Borel counterpart of many other combinatorics), we have a distinct hierarchy of different chain conditions, similar to the classical case.