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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2016
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1608.00146 |
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Table of Contents:
- A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, and it was conjectured that this is the case if the graphs in the sequence are sufficiently sparse. Precisely, two conjectures were proposed: * If a FO-convergent sequence of graphs is residual, that is if for every integer $d$ the maximum relative size of a ball of radius $d$ in the graphs of the sequence tends to zero, then the sequence has a modeling limit. * A monotone class of graphs $\mathcal C$ has the property that every FO-convergent sequence of graphs from $\mathcal C$ has a modeling limit if and only if $\mathcal C$ is nowhere dense, that is if and only if for each integer $p$ there is $N(p)$ such that no graph in $\mathcal C$ contains the $p$th subdivision of a complete graph on $N(p)$ vertices as a subgraph.