On symmetric indivisibility of countable structures

A structure M ≤ N is symmetrically embedded in N if any σ ∈ Aut(M) extends to an automorphism of N . A countable structure M is symmetrically indivisible if for any coloring of M by two colors there exists a monochromatic M ≤ M such that M ∼= M and M is symmetrically embedded in M. We give a model-theoretic proof of the symmetric indivisibility of Rado’s countable random graph [9] and use these new techniques to prove that Q and the generic countable triangle-free graph Γ△ are symmetrically indivisible. Symmetric indivisibility of Q follows from a stronger result, that symmetrically embedded elementary submodels of (Q,≤) are dense. As shown by an anonymous referee, the symmetrically embedded submodels of the random graph are not dense.