{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:37Z","timestamp":1753893817981,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We show that for $n \\geq 3, n\\ne 5$, in any partition of $\\mathcal{P}(n)$, the set of all subsets of $[n]=\\{1,2,\\dots,n\\}$, into $2^{n-2}-1$ parts, some part must contain a triangle \u2014\u00a0three different subsets $A,B,C\\subseteq [n]$ such that $A\\cap B,A\\cap C,B\\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts.\u00a0 We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.<\/jats:p>","DOI":"10.37236\/8385","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T06:45:32Z","timestamp":1578638732000},"source":"Crossref","is-referenced-by-count":0,"title":["Partitioning the Power Set of $[n]$ into $C_k$-free parts"],"prefix":"10.37236","volume":"26","author":[{"given":"Eben","family":"Blaisdell","sequence":"first","affiliation":[]},{"given":"Andr\u00e1s","family":"Gy\u00e1rf\u00e1s","sequence":"additional","affiliation":[]},{"given":"Robert A.","family":"Krueger","sequence":"additional","affiliation":[]},{"given":"Ronen","family":"Wdowinski","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,8,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p38\/7898","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p38\/7898","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:09:12Z","timestamp":1579234152000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i3p38"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,30]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/8385","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,8,30]]},"article-number":"P3.38"}}