{"paper":{"title":"Extremal results on intersection graphs of boxes in $R^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Mart\\'inez-P\\'erez, D. Oliveros, L. Montejano","submitted_at":"2014-12-28T17:30:45Z","abstract_excerpt":"The main purpose of this paper is to study extremal results on the intersection graphs of boxes in $\\R^d$. We calculate exactly the maximal number of intersecting pairs in a family $\\F$ of $n$ boxes in $\\R^d$ with the property that no $k+1$ boxes in $\\F$ have a point in common. This allows us to improve the known bounds for the fractional Helly theorem for boxes. We also use the Fox-Gromov-Lafforgue-Naor-Pach results to derive a fractional Erd\\H{o}s-Stone theorem for semi-algebraic graphs in order to obtain a second proof of the fractional Helly theorem for boxes."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8190","kind":"arxiv","version":2},"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"}