## Topic outline

### Introduction

**This course is now over.**It was held by Raphaël Carroy in the Spring of 2021,- Monday and Friday morning 10.45 am - 12.30 am,

- starting on

**April 12th**and until June 4th.For a total duration of 30 hours (6 CFU).

The course was divided into three parts:

1) Graph and hypergraph dichotomies and (some of) their applications,

2) Topological embeddability on functions,

3) Well-quasi-orders and better-quasi-orders.

### Graph and hypergraph dichotomies

In this first series of lectures, we set the general framework (lecture 1), we prove the G0 dichotomy following Benjamin Miller's methods (lecture 2), and we prove the box-open hypergraph dichotomy (lecture 3). We also discuss some of their applications.

Using the box-open hypergraph dichotomy, Salvatore Scamperti proved the Jayne-Rogers Theorem during the fifteenth (and last) lecture.

**Resources:**The notes for the first two lectures are available as resources, the source material for the second lecture was entirely taken from Benjamin Miller's works, it can be found in this set of notes (notably the proof of the G0 dichotomy) and in this survey article.

The third lecture follows closely the first two sections of this article, joint work with Benjamin Miller and Daniel Soukup.

The first lecture was not recorded (apologies), but the notes for the first lecture cover quite extensively its contents.

The other two lectures for this part were recorded. I apologize for the poor quality of the video, I have changed the recording settings so this inconvenience disappears for further lectures.

### Topological embeddability between functions

In the fourth lecture, we show a proof of the Hurewicz dichotomy from the box-open hypergraph dichotomy, and we introduce the quasi-order of topological embeddability between functions.

`We then prove, in the fifth and sixth lecture, a weak version of the finite basis result theorem for non-Baire-class-one functions on separable metric spaces obtained with B.Miller.``We then discuss the descriptive complexity of topological embeddability between continuous functions defined on Polish 0-dimensional spaces. In the seventh lecture, we discuss possible upper bounds for that quasi-order when the functions have sigma-compact domains.``In the eighth and ninth lecture, we prove that analytic hardness is a lower bound for embeddability between continuous functions when the domain has infinitely many limit points and the range is not discrete. This lower bound is proven in a special case in the eighth lecture, and the general case is discussed in the ninth lecture.`**Resources:**The notes for the fourth lecture, as well as those for the finite basis result in common with Benjamin Miller are available. They cover the fourth, fifth, and sixth lecture.

We then follow rather closely the article that is available under the name

*Topological Embeddability - CPV.*The seventh lecture essentially follows section 3, while the eighth and ninth lectures follow subsections 4.1 and 4.2 respectively.### Well-quasi-orders and better-quasi-orders

This third and last part of the course is dedicated to well-quasi-orders and better-quasi orders.

During the tenth lecture, the definition of better-quasi-orders is motivated, and several definitions are given.

In the eleventh lecture, we give a game-theoretical proof of the Silver-Ellentuck theorem due to the combined works of Kastanas and K.Tanaka.

The twelfth and thirteenth lectures are dedicated to Laver's proof of Fraïssé's conjecture, some strengthening, and consequences.

The other side of the dichotomy concerning topological embeddability on spaces of continuous functions was proven during the fourteenth lecture.

**Resources:**The introduction to better-quasi-orders follows the approach from the article

*WBAndInBetween*, written with Yann Pequignot.Some handwritten notes are given for the game-theoretical proof of the Silver-Ellentuck theorem and the proof of Laver's theorem.

The first half of the fourteenth lecture follows Section 5 of the

*CPV*article given as a resource in the second part of the course.