A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems
Combinatorics
2017-01-09 v1
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.
Cite
@article{arxiv.1701.01631,
title = {A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems},
author = {Christoph Spiegel},
journal= {arXiv preprint arXiv:1701.01631},
year = {2017}
}
Comments
14 pages