{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:42Z","timestamp":1753893822983,"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>Three $k$-dimensional subspaces $A$, $B$, and $C$ of an $n$-dimensional vector space $V$ over a finite field are called a $3$-cluster if $A \\cap B \\cap C = \\{\\mathbf{0}_V\\}$ and yet $\\dim(A+B+C) \\leq 2k$. A special kind of $3$-cluster, which we call a covering triple, consists of subspaces $A,B,C$ such that $A = (A \\cap B )\\oplus (A \\cap C)$. We prove that, for $2 \\leq k \\le n\/2$, the largest size of a covering triple-free family of $k$-dimensional subspaces is the same as the size of the largest such star (a family of subspaces all containing a designated non-zero vector). Moreover, we show that if $k &lt; n\/2$, then stars are the only families achieving this largest size. This in turn implies the same result for $3$-clusters, which gives the vector space-analogue of a theorem of Mubayi for set systems.<\/jats:p>","DOI":"10.37236\/12962","type":"journal-article","created":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T14:33:06Z","timestamp":1747924386000},"source":"Crossref","is-referenced-by-count":0,"title":["3-Cluster-Free Families of Subspaces"],"prefix":"10.37236","volume":"32","author":[{"given":"Gabriel","family":"Currier","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shahriar","family":"Shahriari","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,5,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p31\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T14:33:07Z","timestamp":1747924387000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,23]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/12962","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,5,23]]},"article-number":"P2.31"}}