Borel Colouring Bad Sequences

TL;DR

This paper explores the limitations of constructing Borel graphs with infinite chromatic number using better quasi-orders, revealing that standard methods are insufficient and many known quasi-orders are non-examples.

Contribution

It demonstrates the impossibility of explicit construction of certain Borel graphs via standard methods and highlights the existence of an unexplored class of better quasi-orders.

Findings

Standard methods cannot construct Borel graphs with infinite chromatic number.

Most known better quasi-orders are non-examples for this property.

There exists an unknown class of better quasi-orders with interesting properties.

Abstract

Every better quasi-order codifies a Borel graph that does not contain a copy of the shift graph. It is known that there is a better quasi-order that codes a Borel graph with infinite Borel chromatic number, though one has yet to be explicitly constructed. In this paper, we show that examples cannot be constructed via standard methods. Moreover, we show that most of the known better quasi-orders are non-examples, suggesting there is still a class of better quasi-orders with interesting combinatorial properties who's elements/members still remain unknown.