{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:57Z","timestamp":1753893837840,"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 subset $A$ of $\\mathbb{Z}^n$ is called a weak antichain if it does not contain two elements $x$ and $y$ satisfying $x_i&lt;y_i$ for all $i$. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain $A$, the sum of the sizes of its $(n-1)$-dimensional projections must be at least as large as its size $|A|$. They asked what the smallest possible value of the gap between these two quantities is in terms of\u00a0 $|A|$. We answer this question by giving an explicit weak antichain attaining this minimum for each possible value of $|A|$. In particular, we show that sets of the form $$A_N=\\{x\\in\\mathbb{Z}^n: 0\\leq x_j\\leq N-1 \\textrm{ for all $j$ and } x_i=0\\textrm{ for some $i$}\\}$$ minimise the gap among weak antichains of size $|A_N|$.\r\n\u00a0<\/jats:p>","DOI":"10.37236\/9174","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:45Z","timestamp":1584578685000},"source":"Crossref","is-referenced-by-count":1,"title":["Projections of Antichains"],"prefix":"10.37236","volume":"27","author":[{"given":"Barnab\u00e1s","family":"Janzer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,3,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p54\/8042","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p54\/8042","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:45Z","timestamp":1584578685000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/9174","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,3,6]]},"article-number":"P1.54"}}