{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:04Z","timestamp":1753893844049,"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>Mysteriously, hypergraphs that are the intersection of two matroids behave in some respects almost as well as one matroid. In the present paper we study one such phenomenon\u00a0- the surprising ability of the intersection of two matroids to fairly represent the parts of a given partition of the ground set. For a simplicial complex $\\mathcal{C}$ denote by $\\beta(\\mathcal{C})$ the minimal number of edges from $\\mathcal{C}$ needed to cover the ground set. If $\\mathcal{C}$ is a matroid then for every partition $A_1, \\ldots, A_m$ of the ground set there exists a set $S \\in \\mathcal{C}$ meeting each $A_i$ in at least $\\lfloor \\frac{|A_i|}{\\beta(\\mathcal{C})}\\rfloor$ elements. We conjecture that a slightly weaker result is true for the intersection of two matroids: if $\\mathcal{D}=\\mathcal{P}\\cap \\mathcal{Q}$, where $\\mathcal{P},\\mathcal{Q}$ are matroids on the same ground set $V$ and $\\beta(\\mathcal{P}), \\beta(\\mathcal{Q}) \\le k$, then for every partition $A_1, \\ldots, A_m$ of the ground set there exists a set $S \\in \\mathcal{D}$ meeting each $A_i$ in at least $\\frac{1}{k}|A_i|-1$ elements. We prove that if $m=2$ (meaning that the partition is into two sets) there is a set belonging to $\\mathcal{D}$ meeting each $A_i$ in at least $(\\frac{1}{k}-\\frac{1}{|V|})|A_i|-1$ elements.<\/jats:p>","DOI":"10.37236\/6946","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:44:48Z","timestamp":1578671088000},"source":"Crossref","is-referenced-by-count":0,"title":["Fair Representation in the Intersection of Two Matroids"],"prefix":"10.37236","volume":"24","author":[{"given":"Ron","family":"Aharoni","sequence":"first","affiliation":[]},{"given":"Eli","family":"Berger","sequence":"additional","affiliation":[]},{"given":"Dani","family":"Kotlar","sequence":"additional","affiliation":[]},{"given":"Ran","family":"Ziv","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,10,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p10\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:45:09Z","timestamp":1579236309000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i4p10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,6]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/6946","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,10,6]]},"article-number":"P4.10"}}