Topology, forcing, and graph colourings

TL;DR

This paper introduces forcing notions to demonstrate the non-existence of countable Borel class alpha colourings for certain graphs, and constructs weakly minimal graphs for these colourings.

Contribution

It presents new forcing notions and constructs weakly minimal graphs to analyze Borel class colourings, advancing understanding in descriptive graph theory.

Findings

Forcing notions help show graphs lack countable Borel class alpha colourings.

Constructed graphs are weakly minimal for such colourings.

Provides tools for analyzing graph colourings in descriptive set theory.

Abstract

We introduce a family of forcing notions that are helpful in showing that certain graphs do not have countable colourings of (additive) Borel class alpha. We construct graphs that are ''weakly minimal'' for such colourings.