{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:13Z","timestamp":1753893853882,"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>The critical number $cr(r,n)$ of natural intervals $[r,n]$ was introduced by Herzog, Kaplan and Lev in 2014. The critical number $cr(r,n)$ is the smallest integer $t$ satisfying the following conditions: (i) every sequence of integers $S=\\{r_1=r\\leq r_2\\leq \\dotsb\\leq r_k\\}$ with\u00a0 $r_1+r_2 +\\dotsb +r_k=n$ and $k\\geq t$ has the following property: every integer between $r$ and $n-r$ can be written as a sum of distinct elements of $S$, and (ii) there exists $S$ with $k=t$, which satisfies that property. In this paper we study a variation of the critical number $cr(r,n)$ called the $r$-critical number $rcr(r,n)$.\u00a0 We determine the value of $rcr(r,n)$ for all $r,n$ satisfying $r\\mid n$.<\/jats:p>","DOI":"10.37236\/9835","type":"journal-article","created":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T02:05:37Z","timestamp":1638497137000},"source":"Crossref","is-referenced-by-count":0,"title":["$r$-Critical Numbers of Natural Intervals"],"prefix":"10.37236","volume":"28","author":[{"given":"Marcel","family":"Herzog","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gil","family":"Kaplan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arieh","family":"Lev","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Romina","family":"Zigdon","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,12,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p42\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p42\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T02:05:42Z","timestamp":1638497142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i4p42"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,3]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,10,8]]}},"URL":"https:\/\/doi.org\/10.37236\/9835","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,12,3]]},"article-number":"P4.42"}}