{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:39Z","timestamp":1753893819480,"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>The Galvin problem asks for the minimum size of a family $\\mathcal{F} \\subseteq \\binom {[n]} {n\/2}$ with the property that, for any set $A$ of size $\\frac n 2$, there is a set $S \\in \\mathcal{F}$ which is balanced on $A$, meaning that $|S \\cap A| = |S \\cap \\overline{A}|$. We consider a generalization of this question that comes from a possible approach in complexity theory. In the generalization the required property is, for any $A$, to be able to find $d$ sets from a family $\\mathcal{F} \\subseteq \\binom {[n]} {n\/d}$ that form a partition of $[n]$ and such that each part is balanced on $A$. We construct such families of size polynomial in the parameters $n$ and $d$.<\/jats:p>","DOI":"10.37236\/8432","type":"journal-article","created":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T10:05:00Z","timestamp":1581069900000},"source":"Crossref","is-referenced-by-count":0,"title":["d-Galvin Families"],"prefix":"10.37236","volume":"27","author":[{"given":"Johan","family":"H\u00e5stad","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guillaume","family":"Lagarde","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph","family":"Swernofsky","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,2,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p36\/8021","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p36\/8021","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T10:05:00Z","timestamp":1581069900000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p36"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8432","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,2,7]]},"article-number":"P1.36"}}