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First-order limits, an analytical perspective
In this paper we present a novel approach to graph (and structural) limits
based on model theory and analysis. The role of Stone and Gelfand dualities is
displayed prominently and leads to a general theory, which we believe is
naturally emerging. This approach covers all the particular examples of
structural convergence and it put the whole in new context. As an application,
it leads to new intermediate examples of structural convergence and to a "grand
conjecture" dealing with sparse graphs. We survey the recent developments
Graph limits and exchangeable random graphs
We develop a clear connection between deFinetti's theorem for exchangeable
arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph
limits (work of Lovasz and many coauthors). Along the way, we translate the
graph theory into more classical probability.Comment: 26 page
Limits of Structures and the Example of Tree-Semilattices
The notion of left convergent sequences of graphs introduced by Lov\' asz et
al. (in relation with homomorphism densities for fixed patterns and
Szemer\'edi's regularity lemma) got increasingly studied over the past
years. Recently, Ne\v set\v ril and Ossona de Mendez introduced a general
framework for convergence of sequences of structures. In particular, the
authors introduced the notion of -convergence, which is a natural
generalization of left-convergence. In this paper, we initiate study of
-convergence for structures with functional symbols by focusing on the
particular case of tree semi-lattices. We fully characterize the limit objects
and give an application to the study of left convergence of -partite
cographs, a generalization of cographs
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