{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:65LCIRSIKPVBKBC4AC76JGRQW2","short_pith_number":"pith:65LCIRSI","schema_version":"1.0","canonical_sha256":"f75624464853ea15045c00bfe49a30b6b3840dcc0fd14ab13167703d885b4ea3","source":{"kind":"arxiv","id":"1211.1319","version":1},"attestation_state":"computed","paper":{"title":"Shattering, Graph Orientations, and Connectivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"Laszlo Kozma, Shay Moran","submitted_at":"2012-11-06T17:36:15Z","abstract_excerpt":"We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied concepts of graph theory related to connectivity, combinatorial optimization, forbidden subgraphs, and others.\n  In one direction, we use this connection to derive results in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma. Using this tool we obtain a series of inequalities and equalities related to properties of orientations of a graph"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.1319","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-11-06T17:36:15Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"ed938c2b1fa2b89641dfc07c30c46fab10a021a1f2a285d996f630196955d414","abstract_canon_sha256":"4fa63e253fed63c1dcca041ce2f51be1da22a1fe21fb997321c7cf1bd9c7389d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:21.849284Z","signature_b64":"aC0TyQA/JUIRO2r9IYN4lpq4DOF/yiK1xVX2Hc0dDhavTghjYRPPl/ilfVlwPupav8VHrar3Yq4ukkEtsrkOAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f75624464853ea15045c00bfe49a30b6b3840dcc0fd14ab13167703d885b4ea3","last_reissued_at":"2026-05-18T03:41:21.848905Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:21.848905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shattering, Graph Orientations, and Connectivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"Laszlo Kozma, Shay Moran","submitted_at":"2012-11-06T17:36:15Z","abstract_excerpt":"We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied concepts of graph theory related to connectivity, combinatorial optimization, forbidden subgraphs, and others.\n  In one direction, we use this connection to derive results in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma. Using this tool we obtain a series of inequalities and equalities related to properties of orientations of a graph"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1319","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1211.1319","created_at":"2026-05-18T03:41:21.848974+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.1319v1","created_at":"2026-05-18T03:41:21.848974+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1319","created_at":"2026-05-18T03:41:21.848974+00:00"},{"alias_kind":"pith_short_12","alias_value":"65LCIRSIKPVB","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_16","alias_value":"65LCIRSIKPVBKBC4","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_8","alias_value":"65LCIRSI","created_at":"2026-05-18T12:26:56.085431+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2","json":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2.json","graph_json":"https://pith.science/api/pith-number/65LCIRSIKPVBKBC4AC76JGRQW2/graph.json","events_json":"https://pith.science/api/pith-number/65LCIRSIKPVBKBC4AC76JGRQW2/events.json","paper":"https://pith.science/paper/65LCIRSI"},"agent_actions":{"view_html":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2","download_json":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2.json","view_paper":"https://pith.science/paper/65LCIRSI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.1319&json=true","fetch_graph":"https://pith.science/api/pith-number/65LCIRSIKPVBKBC4AC76JGRQW2/graph.json","fetch_events":"https://pith.science/api/pith-number/65LCIRSIKPVBKBC4AC76JGRQW2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/action/storage_attestation","attest_author":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/action/author_attestation","sign_citation":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/action/citation_signature","submit_replication":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/action/replication_record"}},"created_at":"2026-05-18T03:41:21.848974+00:00","updated_at":"2026-05-18T03:41:21.848974+00:00"}