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The Generic Bipartite Graphs of Diameter 3: Their Ages and their Almost Sure Theories

In an effort to find more examples of amalgamation classes whose almost sure theories are the same as their generic theories as well as amalgamation classes whose almost sure theories are different from their generic theories, we address our attention to two new cases: the bipartite diameter 3 metrically homogeneous graphs of generic type. These graphs were identified by Cherlin, and are determined by certain forbidden configurations. In this paper, we explicitly identify and enumerate their ages, for which we then establish both unlabeled and labeled 0-1 laws. Finally, we show that for one of these bipartite graphs the almost sure theory matches its generic theory, and for the other bipartite graph it does not.