{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:15Z","timestamp":1753893855398,"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>A graph $G$ with vertex set $V$ is said to be $n$-existentially closed if, for every $S \\subset V$ with $|S|=n$ and every $T \\subseteq S$, there exists a vertex $x \\in V-S$ such that $x$ is adjacent to each vertex of $T$ but is adjacent to no vertex of $S-T$.  Given a combinatorial design ${\\cal D}$ with block set ${\\cal B}$, its block-intersection graph $G_{{\\cal D}}$ is the graph having vertex set ${\\cal B}$ such that two vertices $b_1$ and $b_2$ are adjacent if and only if $b_1$ and $b_2$ have non-empty intersection. In this paper we study BIBDs (balanced incomplete block designs) and when their block-intersection graphs are $n$-existentially closed.  We characterise the BIBDs with block size $k \\geq 3$ and index $\\lambda=1$ that have 2-e.c. block-intersection graphs and establish bounds on the parameters of BIBDs with index $\\lambda=1$ that are $n$-e.c. where $n \\geq 3$.  For $\\lambda \\geq 2$ and $n \\geq 2$, we prove that only simple $\\lambda$-fold designs can have $n$-e.c. block-intersection graphs.  In the case of $\\lambda$-fold triple systems we show that $n \\geq 3$ is impossible, and we determine which 2-fold triple systems (i.e., BIBDs with $k=3$ and $\\lambda=2$) have 2-e.c. block-intersection graphs.<\/jats:p>","DOI":"10.37236\/988","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:48:15Z","timestamp":1578718095000},"source":"Crossref","is-referenced-by-count":5,"title":["Existentially Closed BIBD Block-Intersection Graphs"],"prefix":"10.37236","volume":"14","author":[{"given":"Neil A.","family":"McKay","sequence":"first","affiliation":[]},{"given":"David A.","family":"Pike","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,10,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r70\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r70\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:01:04Z","timestamp":1579320064000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1r70"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,10,16]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/988","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2007,10,16]]},"article-number":"R70"}}