{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:45:26Z","timestamp":1759063526466,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We say that a set system $\\mathcal{F}\\subseteq 2^{[n]}$ shatters a\u00a0given set $S\\subseteq [n]$ if $2^S=\\{F~\\cap~S ~:~F~\\in~\\mathcal{F}\\}$. The Sauer inequality states that in\u00a0general, a set system $\\mathcal{F}$ shatters at least\u00a0$|\\mathcal{F}|$ sets. Here we concentrate on the case of equality.\u00a0A set system is called shattering-extremal if it shatters exactly\u00a0$|\\mathcal{F}|$ sets. In this paper we characterize\u00a0shattering-extremal set systems of Vapnik-Chervonenkis dimension\u00a0$2$ in terms of their inclusion graphs, and as a corollary we\u00a0answer an open question about leaving out elements from\u00a0shattering-extremal set systems in the case of families of\u00a0Vapnik-Chervonenkis dimension $2$.<\/jats:p>","DOI":"10.37236\/4548","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:52:40Z","timestamp":1578703960000},"source":"Crossref","is-referenced-by-count":5,"title":["Shattering-Extremal Set Systems of VC Dimension at most 2"],"prefix":"10.37236","volume":"21","author":[{"given":"Tam\u00e1s","family":"M\u00e9sz\u00e1ros","sequence":"first","affiliation":[]},{"given":"Lajos","family":"R\u00f3nyai","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2014,11,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i4p30\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i4p30\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:40:39Z","timestamp":1579257639000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i4p30"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,6]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4548","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,11,6]]},"article-number":"P4.30"}}