In the 1980s, Jarik Nešetřil investigated Ramsey classes: these are classes of structures over a fixed relational language satisfying the condition that, for any structures A,B, there exists C such that, if the embeddings A→C are coloured red and blue, there is an embedding B→C such that the embeddings of A into the image of the embedding of B are monochromatic. (This generalises the classical Ramsey theorem.)

Among his discoveries was the fact that the structures in a nontrivial Ramsey class form a Fraïssé class (and so have a countable homogeneous Fraïssé limit), and are rigid (they have trivial automorphism group).

This was subsequently explained by the celebrated theorem of Kechris, Pestov and Todorčević, which uses topological dynamics (specifically, the automorphism group of the Fraïssé limit is extremely amenable) to show that this group has a total order as a reduct, and hence the structures in the class are totally ordered (and hence rigid).

Early in this story, I observed that there exist Fraïssé classes of rigid structures which have no “natural” total order, and wondered whether they could be Ramsey classes, and (after the KPT theorem showed that they were not) whether one could find explicit failures of the Ramsey property.

Siavash Lashkarighouchani and I have just done this. The paper is on the arXiv, at 2512.05684.

About Peter Cameron

I count all the things that need to be counted.