{"paper":{"title":"Bisecting and D-secting families for set systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Niranjan Balachandran, Rogers Mathew, Sudebkumar Prasant Pal, Tapas Kumar Mishra","submitted_at":"2016-04-06T04:48:20Z","abstract_excerpt":"Let $n$ be any positive integer and $\\mathcal{F}$ be a family of subsets of $[n]$.\n  A family $\\mathcal{F}'$ is said to be $D$-\\emph{secting} for $\\mathcal{F}$ if for every $A \\in \\mathcal{F}$, there exists a subset $A' \\in \\mathcal{F}'$ such that $|A \\cap A'| - |A \\cap ([n] \\setminus A')|=i$, where $i \\in D$, $D \\subseteq \\{-n,-n+1,\\ldots,0,\\ldots,n\\}$.\n  A $D$-\\emph{secting} family $\\mathcal{F}'$ of $\\mathcal{F}$, where $D=\\{-1,0,1\\}$, is a \\emph{bisecting} family ensuring the existence of a subset $A' \\in \\mathcal{F}'$ such that $|A \\cap A'| \\in \\{\\lceil \\frac{|A|}{2}\\rceil,\\lfloor \\frac{|A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01482","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}