On 0, 1-laws and asymptotics of definable sets in geometric Fräıssé classes

We examine one consequence for the generic theory TC of a geometric Fräıssé class C when C has the 0, 1-law for first-order logic with convergence to TC itself. We show that in this scenario, if the asymptotic probability measure in play is not terribly exotic, then C is “very close” to being a 1-dimensional asymptotic class—so that TC is supersimple of finite SU -rank. Introduction. In much of the existing work on 0, 1-laws for Fräıssé classes C, researchers have focused almost entirely on the most ordinary of asymptotic probability measures μ = (μN )N—namely, μN is the uniform probability measure on members of C with universe N = {0, 1, . . . , N − 1}. When there is a 0, 1-law relative to such μ with Th(μ) = TC ( 1), one very often finds that TC was already a very special sort of theory. For example, if G is the class of all finite graphs, then TG, the theory of the random graph, is supersimple of SU -rank 1, and one finds something similar when C is the class of finite bipartite graphs, the class of finite partial orders of rank 2, finite directed trees of height 2, and so on. There is a sense that if a class C has the 0, 1-law in a way similar to G, then geometrically speaking, TC is very much like TG. This discussion requires an answer to the question, “What does it mean for C to have the 0, 1-law in a way similar to G?” In this paper, we answer this question for geometric Fräıssé classes (i.e. TC is a geometric theory) by focusing on (i) the conditional independence properties of the asymptotic probability measure μ, and (ii) the requirement that Th(μ) = TC. We find 2010 Mathematics Subject Classification: 03C13, 03C15, 03C45, 05A16.