{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:58Z","timestamp":1753893838249,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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\u00e9meredi-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\u00f6dl, Ruci\u0144ski 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\u00fazsa as well as Ru\u00e9 et al.<\/jats:p>","DOI":"10.37236\/6730","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:49:44Z","timestamp":1578653384000},"source":"Crossref","is-referenced-by-count":4,"title":["A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems"],"prefix":"10.37236","volume":"24","author":[{"given":"Christoph","family":"Spiegel","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,8,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p38\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p38\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:49:48Z","timestamp":1579218588000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p38"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,25]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6730","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,8,25]]},"article-number":"P3.38"}}