{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QB7XBWL52OJV4OWRVPRQEE44KB","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":"85f979f577d91ac80dfa38a4560e58c94ed380cf2e8868971938b4128c709120","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-07T09:20:26Z","title_canon_sha256":"8ede9a824188fd6ff07988fb9771497cb10de670e6b7c72a7e0e3d4566027d84"},"schema_version":"1.0","source":{"id":"1806.02574","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.02574","created_at":"2026-07-05T00:15:00Z"},{"alias_kind":"arxiv_version","alias_value":"1806.02574v5","created_at":"2026-07-05T00:15:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.02574","created_at":"2026-07-05T00:15:00Z"},{"alias_kind":"pith_short_12","alias_value":"QB7XBWL52OJV","created_at":"2026-07-05T00:15:00Z"},{"alias_kind":"pith_short_16","alias_value":"QB7XBWL52OJV4OWR","created_at":"2026-07-05T00:15:00Z"},{"alias_kind":"pith_short_8","alias_value":"QB7XBWL5","created_at":"2026-07-05T00:15:00Z"}],"graph_snapshots":[{"event_id":"sha256:4602175badb44a883c9af7854315f15af5f0a1a4134d7a32e238d90217ccb532","target":"graph","created_at":"2026-07-05T00:15:00Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1806.02574/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the $d$-dimensional rectilinear drawings of the complete $d$-uniform hypergraph $K_{2d}^d$. Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove that there exist $\\Omega \\left(2^d\\right)$ crossing pairs of hyperedges in such a drawing of $K_{2d}^d$. We improve this lower bound by showing that there exist $\\Omega \\left(2^d \\sqrt{ d}\\right)$ crossing pairs of hyperedges in a $d$-dimensional rectilinear drawing of $K_{2d}^d$. We also prove the following results.\n  1. There are $\\Omega \\left(2^d {d^{3/2}}","authors_text":"Rahul Gangopadhyay, Saswata Shannigrahi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-07T09:20:26Z","title":"$k$-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.02574","kind":"arxiv","version":5},"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":"sha256:3d8e570da882f72e83eb6cf22e9e82816ac83892b5f521b44c0f3c8b9d80a5a2","target":"record","created_at":"2026-07-05T00:15:00Z","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":"85f979f577d91ac80dfa38a4560e58c94ed380cf2e8868971938b4128c709120","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-07T09:20:26Z","title_canon_sha256":"8ede9a824188fd6ff07988fb9771497cb10de670e6b7c72a7e0e3d4566027d84"},"schema_version":"1.0","source":{"id":"1806.02574","kind":"arxiv","version":5}},"canonical_sha256":"807f70d97dd3935e3ad1abe302139c50593d717adf5f50473f03ca673bfebb4b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"807f70d97dd3935e3ad1abe302139c50593d717adf5f50473f03ca673bfebb4b","first_computed_at":"2026-07-05T00:15:00.000550Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:15:00.000550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"avqIZE5Sq0I4puOVmctH7XFzytJA90KiqTa+urHFEhS5vBCmfD4OM9f2R0BNWR0jSVgWZpM7oiQfR5bywF/8AA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:15:00.000931Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.02574","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d8e570da882f72e83eb6cf22e9e82816ac83892b5f521b44c0f3c8b9d80a5a2","sha256:4602175badb44a883c9af7854315f15af5f0a1a4134d7a32e238d90217ccb532"],"state_sha256":"bf72cd5a20f9c7107ddb5124f662bb8d7db5294fb6eb3ec5c56fc7f3ee986df7"}