{"paper":{"title":"Balanced partitions of 3-colored geometric sets in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Carlos Seara, Dolores Lara, Ferran Hurtado, Jorge Urrutia, Kevin Verbeek, Matias Korman, Mikio Kano, Rodrigo I. Silveira, Sergey Bereg","submitted_at":"2017-08-21T02:27:50Z","abstract_excerpt":"Let $S$ be a finite set of geometric objects partitioned into classes or \\emph{colors}. A subset $S'\\subseteq S$ is said to be \\emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several problems on partitioning $3$-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are $2m$ lines of each color, there is a segment intercepting $m$ lines of each color. (b) Given $n$ red points"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.06062","kind":"arxiv","version":1},"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"}