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arXiv · 1701.01631

A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

Christoph Spiegel
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Abstract

We study the thresholds for the property of containing a solution to a linear homogeneous system in random sets. We expand a previous sparse Sz\'emeredi-type result of Schacht to the broadest class of matrices possible. We also provide a shorter proof of a sparse Rado result of Friedgut, R\"odl, Ruci\'nski and Schacht based on a hypergraph container approach due to Nenadov and Steger. Lastly we further extend these results to include some solutions with repeated entries using a notion of non-trivial solutions due to R\'uzsa as well as Ru\'e et al.

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Christoph Spiegel. 2017-01-06. A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems. https://arxiv.org/abs/1701.01631

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