Did I prove that?

Two interesting and contrasting experiences in two days at the Durham symposium (about which you will hear lots more later). I should begin by saying that these were both during excellent talks.

Yesterday, Rehana Patel gave the first of three talks by her and her collaborators Nate Ackerman and Cameron Freer about ergodic measures concentrated on isomorphism classes of first-order structures, or on first-order sentences. I have described before the content of their first result, which is that there is an exchangeable measure concentrated on the isomorphism type of a first-order structure M if and only if definable closure in M is trivial (which means, in permutation group terms, that the stabiliser of a finite tuple of points in the automorphism group of M fixes no additional points). One new result is that the number of such measures which are ergodic is 0, 1, or the cardinality of the continuum. We know when the case 0 occurs; the case 1 occurs if and only if the automorphism group is highly homogeneous, that is, acts transitively on unordered n-element sets for every natural number n. She quoted my theorem from 1976 (my first result on infinite permutation groups) describing such groups; this shows that there are only five such structures, the countable dense linear order without endpoints, its derived betweenness, circular order, and betweenness relations, and a set without structure. Most people who quote this nowdays say that I found all the reducts of the linear order, that is, all closed overgroups of its automorphism group; I have somehow become a pioneer in the study of reducts. When I proved the theorem, I didn’t even know what a reduct was! In fact my theorem, as stated by Rehana, is more general.

The second occurred today in Christian Rosendal’s talk on the coarse geometric structure of automorphism groups. He attributed to me the theorem, or observation, that any countably categorical structure is quasi-isometric to a point. I was a bit taken aback; the first thought that crossed my mind is that this had something to do with the fact that a countably categorical structure M has the property that any structure N younger than M (that is, all finite substructures of N are embeddable in M) is itself embeddable in M: an application of König’s Infinity Lemma. However, it turned out that this followed from the statement that, for any finite tuple a of elements of M, the stabiliser of a in the automorphism group has only finitely many double cosets, or in other words, the action of the automorphism group on the orbit of a has only finitely many orbits on pairs; a trivial consequence of the Ryll-Nardzewski theorem. (If I did make this comment somewhere, it was certainly before I knew what a quasi-isometry is.)