{"paper":{"title":"Simplifying inclusion-exclusion formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ji\\v{r}\\'i Matou\\v{s}ek, Martin Tancer, Pavel Pat\\'ak, Xavier Goaoc, Zuzana Safernov\\'a","submitted_at":"2012-07-11T10:53:03Z","abstract_excerpt":"Let $\\mathcal{F}=\\{F_1,F_2, \\ldots,F_n\\}$ be a family of $n$ sets on a ground set $S$, such as a family of balls in $\\mathbb{R}^d$. For every finite measure $\\mu$ on $S$, such that the sets of $\\mathcal{F}$ are measurable, the classical inclusion-exclusion formula asserts that $\\mu(F_1\\cup F_2\\cup\\cdots\\cup F_n)=\\sum_{I:\\emptyset\\ne I\\subseteq[n]} (-1)^{|I|+1}\\mu\\Bigl(\\bigcap_{i\\in I} F_i\\Bigr)$; that is, the measure of the union is expressed using measures of various intersections. The number of terms in this formula is exponential in $n$, and a significant amount of research, originating in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2591","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"}