{"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":1753893838448,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \\times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) \\neq O(n)$ but $ex(n, P') = O(n)$ for every proper subpattern $P'$ of $P$. We prove that the ratio between the length and width of any minimally non-linear 0-1 matrix is at most $4$, and that a minimally non-linear 0-1 matrix with $k$ rows has at most $5k-3$ ones. We also obtain an upper bound on the number of minimally non-linear 0-1 matrices with $k$ rows.In addition, we prove corresponding bounds for minimally non-linear ordered graphs. The minimal non-linearity that we investigate for ordered graphs is for the extremal function $ex_{&lt;}(n, G)$, which is the maximum possible number of edges in any ordered graph on $n$ vertices with no ordered subgraph isomorphic to $G$.<\/jats:p>","DOI":"10.37236\/6735","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:42:28Z","timestamp":1578652948000},"source":"Crossref","is-referenced-by-count":2,"title":["Bounds on Parameters of Minimally Nonlinear Patterns"],"prefix":"10.37236","volume":"25","author":[{"given":"P. A.","family":"CrowdMath","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,1,12]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p5\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:41:35Z","timestamp":1579218095000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i1p5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,12]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/6735","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,1,12]]},"article-number":"P1.5"}}