{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T03:18:15Z","timestamp":1761621495021,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>An intersection graph defines an adjacency relation between subsets $S_1,\\dots, S_n$ of a finite set $W=\\{w_1,\\dots, w_m\\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\\mathbb{P}(S_i=A)=P(|A|){\\binom{m}{|A|}}^{-1}$, $A\\subseteq W$, where $P$ is a probability on $\\{0, 1, \\dots, m\\}$, we obtain the random intersection graph $G=G(n,m,P)$.\u00a0We establish\u00a0 the asymptotic order of the clique number $\\omega(G)$ of\u00a0 a sparse random intersection graph as $n,m\\to+\\infty$. For $m = \\Theta(n)$ we show that the maximum clique is of size $(1-\\alpha\/2)^{-\\alpha\/2}n^{1-\\alpha\/2}(\\ln n)^{-\\alpha\/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\\alpha \\in (1,2)$. It is of size $\\frac {\\ln n} {\\ln \\ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\\alpha&gt;2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\\omega(G) (1-o_P(1))$.<\/jats:p>","DOI":"10.37236\/6233","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:58:16Z","timestamp":1578671896000},"source":"Crossref","is-referenced-by-count":6,"title":["Large Cliques in Sparse Random Intersection Graphs"],"prefix":"10.37236","volume":"24","author":[{"given":"Mindaugas","family":"Bloznelis","sequence":"first","affiliation":[]},{"given":"Valentas","family":"Kurauskas","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,4,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p5\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:02:29Z","timestamp":1579237349000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i2p5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,13]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/6233","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,4,13]]},"article-number":"P2.5"}}