A well quasi-order (wqo) is a well-founded quasi-order which contains no infinite antichain. The theory of wqos has applications in many contexts and consists essentially of developing tools in order to show that a certain quasi-order suspected to be wqo is indeed so. This theory exhibits a curious and interesting phenomenon: to prove that a certain quasi-order is wqo, it may very well be easier to show that it enjoys a much stronger property. This observation may be seen as a motivation for considering the complicated but ingenious concept of better-quasi-order (bqo) invented by Nash-Williams in 1965.

After a motivated introduction to the concept of bqo, I will sketch the proof of a conjecture made by Pouzet in 1978 which states that any wqo whose ideal completion remainder is bqo is actually bqo. The proof relies on a result with both a combinatorial and a topological flavour concerning maps from a front into a compact metric space.

This is joint work with Raphaël Carroy.