Properties of the automorphism group and a probabilistic construction of a class of countable labeled structures

For a class of countably infinite ultrahomogeneous structures that generalize edge-colored graphs we provide a probabilistic construction. Also, we give fairly general criteria for the automorphism group of such structures to have the small index property and strong uncountable cofinality, thus generalizing some results of Solecki, Rosendal, and several other authors.