{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:43:09Z","timestamp":1787323389587,"version":"build-2736575974"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>Mubayi's conjecture states that if $\\mathcal{F}$ is a family of $k$-sized subsets of $[n] = \\{1,\\ldots,n\\}$ which, for $k \\geq d \\geq 3$ and $n \\geq \\frac{dk}{d-1}$, satisfies $A_1 \\cap\\cdots\\cap A_d \\neq \\emptyset$ whenever $|A_1 \\cup\\cdots\\cup A_d|~\\leq~2k$ for all distinct sets $A_1,\\ldots,A_d \\in\\mathcal{F}$, then $|\\mathcal{F}|\\leq \\binom{n-1}{k-1}$, with equality occurring only if $\\mathcal{F}$ is the family of all $k$-sized subsets containing some fixed element. This paper proves that Mubayi's conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between what we call $(i,j)$-unstable families and $(j,i)$-unstable families. Generalizing previous intersecting conditions, we introduce the $(d,s,t)$-conditionally intersecting condition for families of sets and prove general results thereon. We prove fundamental theorems on two $(d,s)$-conditionally intersecting families that generalize previous intersecting families, and we pose an extension of a previous conjecture by Frankl and F\u00fcredi. Finally, we generalize a classical result by Erd\u00f6s, Ko, and Rado by proving tight upper bounds on the size of $(2,s)$-conditionally intersecting families $\\mathcal{F}\\subseteq 2^{[n]}$ and by characterizing the families that attain these bounds. We extend this theorem for sufficiently large $n$ to families $\\mathcal{F}\\subseteq 2^{[n]}$ whose members have at most a fixed size $u$.<\/jats:p>","DOI":"10.1137\/18m1166250","type":"journal-article","created":{"date-parts":[[2018,9,27]],"date-time":"2018-09-27T10:05:25Z","timestamp":1538042725000},"page":"2361-2380","source":"Crossref","is-referenced-by-count":2,"title":["On Mubayi's Conjecture and Conditionally Intersecting Sets"],"prefix":"10.1137","volume":"32","author":[{"given":"Adam","family":"Mammoliti","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Britz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,9,27]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1996.0092"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1006\/eujc.1995.0092"},{"key":"atypb3","doi-asserted-by":"crossref","first-page":"8","DOI":"10.37236\/2693","volume":"20","author":"Brouwer A. 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Lifshitz,\n                      On Set Systems without a Simplex-Cluster and the Junta Method\n                      , preprint,https:\/\/arxiv.org\/abs\/1804.01026, 2018."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2005.04.001"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2006.11.013"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2008.10.004"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548309009705"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1166250","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:59:45Z","timestamp":1787320785000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1166250"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/18M1166250"],"URL":"https:\/\/doi.org\/10.1137\/18m1166250","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}