• Corpus ID: 226281389

# Smooth approximations and CSPs over finitely bounded homogeneous structures

@article{Mottet2020SmoothAA,
title={Smooth approximations and CSPs over finitely bounded homogeneous structures},
author={Antoine Mottet and Michael Pinsker},
journal={ArXiv},
year={2020},
volume={abs/2011.03978}
}
• Published 8 November 2020
• Mathematics, Computer Science
• ArXiv
We develop the novel machinery of smooth approximations, and apply it to confirm the CSP dichotomy conjecture for first-order reducts of the random tournament, various homogeneous graphs including the random graph, and for expansions of the order of the rationals. Apart from obtaining these dichotomy results, we show how our new proof technique allows to unify and significantly simplify the previous results from the literature. For all but the last structure, we moreover characterize those CSPs…
3 Citations

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• 2017
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