One of the numerous strengthenings of Ramsey's theorem is due to Erdös and Rado, who analyzed what partition properties can be obtained on m-subsets of the naturals when colorings are not necessarily finite. Large monochromatic sets may not appear in that case, but there is a finite list of behaviors, called "canonical", to which every coloring reduces. The purpose of this talk will be to remind certain not so well-known analogous theorems of the same flavor that were obtained by Prömel in the eighties for various classes of structures (like graphs or hypergraphs), to show how such theorems can in fact be deduced in the more general setting of Fraïssé classes, and to present how those theorems reflect at the level of the automorphism group of the

corresponding limits.