{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:29Z","timestamp":1753893809439,"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>For a subset $S$ of a finite ordered set $P$, let $$S\\!\\uparrow\\;=\\{x\\in P:x\\ge s \\hbox{ for some }s\\in S\\}\\quad\\hbox{and} \\quad S\\!\\downarrow\\;=\\{x\\in P:x\\le s \\hbox{ for some }s\\in S\\}.$$ For a maximal antichain $A$ of $P$, let $$s(A)=\\max_{A=U\\cup D}{|U\\!\\uparrow|+|D\\!\\downarrow|\\over|P|}\\ ,$$ the maximum taken over all partitions $U\\cup D$ of $A$, and $$s_k(P)=\\min_{A\\in {\\cal A}(P),|A|=k}s(A)$$ where we assume $P$ contains at least one maximal antichain of $k$ elements. Finally, for a class ${\\cal C}$ of finite  ordered sets, we define  $$s_k({\\cal C})=\\inf_{P\\in {\\cal C}}s_k(P).$$ Thus $s_k({\\cal C})$ is the greatest  proportion $r$ satisfying: every $k$-element maximal antichain of a member  $P$ of ${\\cal C}$ can be \"split\" into sets $U$ and $D$ so that  $U\\!\\uparrow\\cup\\; D\\!\\downarrow$ contains at least $r|P|$ elements.  In this paper we determine $s_k({\\cal G}_k)$ for all $k\\ge 1$, where  ${\\cal G}_k=\\{{\\bf k}\\times{\\bf n}:n\\ge k\\}$ is the family of all $k$ by $n$ \"grids\".<\/jats:p>","DOI":"10.37236\/1914","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:18:12Z","timestamp":1578719892000},"source":"Crossref","is-referenced-by-count":0,"title":["Splitting Numbers of Grids"],"prefix":"10.37236","volume":"12","author":[{"given":"Dwight","family":"Duffus","sequence":"first","affiliation":[]},{"given":"Bill","family":"Sands","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,4,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:50:11Z","timestamp":1579323011000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,4,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1914","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,4,13]]},"article-number":"R17"}}