On compositions of symmetrically and elementarily indivisible structures
Ben Gurion University of the Negev
A structure M in a first order language L is indivisible if for every colouring of its universe in two colours, there is a monochromatic substructure M’ of M such that M’ is isomorphic to M. Additionally, we say that M is symmetrically indivisible if M’ can be chosen to be symmetrically embedded in M (That is, every automorphism of M’ can be can be extended to an automorphism of M}), and that M is elementarily indivisible if M’ can be chosen to be an elementary substructure.
The notion of indivisibility is a long-studied subject. We will present these strengthenings of the notion,
examples and some basic properties. in  several questions regarding these new notions arose: If M is symmetrically indivisible or all of its reducts to a sublanguage symmetrically indivisible? Is an elementarily indivisible structure necessarily homogeneous? Does elementary indivisibility imply symmetric indivisibility?
We will define a new “product” of structures, generalising the notions of lexicographic order and lexicographic product of graphs, which preserves indivisibility properties and use it to answer the questions above.
 Assaf Hasson, Menachem Kojman and Alf Onshuus, On symmetric indivisibility of countable structures, Model Theoretic Methods in Finite Combinatorics, AMS, 2011, pp.417–452.