Counting siblings of countable relational structures

Robert Woodrow, University of Calgary. Part of the Logic Seminar Series.

If R and S are two relational structures for the same language, we say that they are siblings when R is isomorphic to an induced subrelational structure of S and vice-versa.  For a given relational countably infinite structure R, sib(R) counts the number of siblings of R up to isomorphism.  There is a conjecture of Thomassé that sib(R) is either one or infinite, and a finer version of this that it is 1, aleph_0 or the continuum.  The problem appears to be very difficult for general structures, even for undirected graphs that are trees.  After setting the situation, we will describe some positive results of Laflamme, Pouzet, Sauer and myself for R which are aleph_0 categorical (whose automorphism group is oligomorphic).

Robert Woodrow, University of Calgary.