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

## TL;DR

This paper investigates thresholds for solutions to linear homogeneous systems in random sets, extending sparse Szemerédi and Rado results to broader classes and solutions with repeated entries using hypergraph container methods.

## Contribution

It broadens the class of matrices for which sparse extremal results hold and provides a shorter proof of a sparse Rado theorem, also including solutions with repeated entries.

## Key findings

- Extended sparse Szemerédi-type results to broader matrix classes
- Provided a shorter proof of sparse Rado results using hypergraph containers
- Included solutions with repeated entries in the analysis

## 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.