{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MHF3WBM5U4AIQA436ZJFXLYBVC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"17c0c62131dab3f6bcfb9d20050bfd2f5ead2626780c04563acbb6326b919a80","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-11T10:53:03Z","title_canon_sha256":"af8c7e93e054834146a4d93779813c1872b9a03b329cfd88470228bedffb5548"},"schema_version":"1.0","source":{"id":"1207.2591","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2591","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2591v2","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2591","created_at":"2026-05-18T02:54:05Z"},{"alias_kind":"pith_short_12","alias_value":"MHF3WBM5U4AI","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MHF3WBM5U4AIQA43","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MHF3WBM5","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:f0d41ffa065e7d3943dda9835603e16f5adaaedd709c9528c65f0336a3acea7d","target":"graph","created_at":"2026-05-18T02:54:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Ji\\v{r}\\'i Matou\\v{s}ek, Martin Tancer, Pavel Pat\\'ak, Xavier Goaoc, Zuzana Safernov\\'a","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-11T10:53:03Z","title":"Simplifying inclusion-exclusion formulas"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2591","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:67ef6c7b55ce93a308ef8d51ba53274630948cac7e42362b8fca8950c147c39e","target":"record","created_at":"2026-05-18T02:54:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"17c0c62131dab3f6bcfb9d20050bfd2f5ead2626780c04563acbb6326b919a80","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-11T10:53:03Z","title_canon_sha256":"af8c7e93e054834146a4d93779813c1872b9a03b329cfd88470228bedffb5548"},"schema_version":"1.0","source":{"id":"1207.2591","kind":"arxiv","version":2}},"canonical_sha256":"61cbbb059da70088039bf6525baf01a8902b88d53110fc3b744001932d5de8d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61cbbb059da70088039bf6525baf01a8902b88d53110fc3b744001932d5de8d7","first_computed_at":"2026-05-18T02:54:05.377290Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:05.377290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CIZ5I1Mp+T12R+ZalE9Yh3v4+GcQWOd/14NNdwfMjWybdLoQGcz+eQmqej93b82m0CLCW1W+89lyDvqCocwqCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:05.377960Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.2591","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67ef6c7b55ce93a308ef8d51ba53274630948cac7e42362b8fca8950c147c39e","sha256:f0d41ffa065e7d3943dda9835603e16f5adaaedd709c9528c65f0336a3acea7d"],"state_sha256":"b7ebc9954c8efd9701d513e632f047b7ae8f3ea8dc7d52f0e8dd32c5fc227e8a"}