{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:50Z","timestamp":1753893830854,"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>Let $k\\ge d\\ge 3$ be fixed. Let $\\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$:\r\n(a)\u00a0$(d,2k+d-3)$-conditionally intersecting families with $n\\ge 3k^5$;\r\n(b) $(k,2k)$-conditionally intersecting families with $n\\ge k^2\/(k-1)$;\r\n(c) Nonintersecting $(3,2k)$-conditionally intersecting families with $n\\ge 3k\\binom{2k}{k}$.\r\nOur results for $(c)$ confirms a conjecture of Mammoliti and Britz for the case $d=3$.<\/jats:p>","DOI":"10.37236\/8894","type":"journal-article","created":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:19:57Z","timestamp":1590718797000},"source":"Crossref","is-referenced-by-count":0,"title":["Structural Results for Conditionally Intersecting Families and some Applications"],"prefix":"10.37236","volume":"27","author":[{"given":"Xizhi","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,5,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p33\/8089","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p33\/8089","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:19:57Z","timestamp":1590718797000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,29]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/8894","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,5,29]]},"article-number":"P2.33"}}