{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:00Z","timestamp":1753893840993,"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>We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Tur\u00e1n numbers of complete, balanced bipartite graphs.  Namely, for fixed $r \\ge 5$, we show that ex$(n, K_{r,r}) = \\Omega(n^{2-2\/(r+1)}(\\ln\\ln n)^{1\/(r^2-1)})$. This improves an old lower bound of Erd\u0151s and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.<\/jats:p>","DOI":"10.37236\/93","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:36:52Z","timestamp":1578717412000},"source":"Crossref","is-referenced-by-count":8,"title":["Lower Bounds for the Size of Random Maximal $H$-Free Graphs"],"prefix":"10.37236","volume":"16","author":[{"given":"Guy","family":"Wolfovitz","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,1,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T03:14:44Z","timestamp":1579317284000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/93","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,1,7]]},"article-number":"R4"}}