Symmetrized Induced Ramsey Theory

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors). We consider vertex and edge colorings, and extensions of partial isomorphisms in the set of all partial isomorphisms between singletons as considered by Babai and Sós (European J Combin 6(2):101-114, 1985), the set of all finite partial isomorphisms as considered by Hrushovski (Combinatorica 12(4):411-416, 1992), Herwig (Combinatorica 15:365-371, 1995) and Herwig-Lascar (Trans Amer Math Soc 5:1985-2021, 2000), and the set of all total isomorphisms. We observe that every finite graph embeds into a finite vertex transitive graph by a so called bi-embedding, an embedding that is compatible with a monomorphism between the corresponding automorphism groups. We also show that every countable graph bi-embeds into Rado's universal countable graph Γ.