Abstract
Mathematicians imagine a myriad of objects, most of them infinite, and inevitably followed by an infinite suite. One way to understand them consists of arranging them, ordering them. This act of organising objects amounts to considering an instance of the very general mathematical notion of a quasi-order. Well-founded quasi-orders play a crucial role in many areas of mathematics, and so do the stronger notions of well-quasi-orders and better-quasi-orders.

This first talk will be an introduction to better-quasi-orders. We will see that their definition hinges on a particularly interesting Borel graph: the shift graph. Its chromatic number is 2, but its Borel chromatic number is infinite.