{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:65LCIRSIKPVBKBC4AC76JGRQW2","short_pith_number":"pith:65LCIRSI","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"},"canonical_sha256":"f75624464853ea15045c00bfe49a30b6b3840dcc0fd14ab13167703d885b4ea3","source":{"kind":"arxiv","id":"1211.1319","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1319","created_at":"2026-05-18T03:41:21Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1319v1","created_at":"2026-05-18T03:41:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1319","created_at":"2026-05-18T03:41:21Z"},{"alias_kind":"pith_short_12","alias_value":"65LCIRSIKPVB","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"65LCIRSIKPVBKBC4","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"65LCIRSI","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:65LCIRSIKPVBKBC4AC76JGRQW2","target":"record","payload":{"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"},"canonical_sha256":"f75624464853ea15045c00bfe49a30b6b3840dcc0fd14ab13167703d885b4ea3","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"},"source_kind":"arxiv","source_id":"1211.1319","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"obSMZ7dyRRpAzAT7bpl4nRJgqi96TYJKEBDboe6Bp6fuoNHgt9frnFUanAatvY4tuARAg7S9l0CRu1CXhkxNAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T01:33:47.152090Z"},"content_sha256":"b9c5d98d8e30920e59b04553ffe48ff532ba8a9740f94b7b216b46bca36351a9","schema_version":"1.0","event_id":"sha256:b9c5d98d8e30920e59b04553ffe48ff532ba8a9740f94b7b216b46bca36351a9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:65LCIRSIKPVBKBC4AC76JGRQW2","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M6tlSajH6wafbV5vSj1IAAYvDwjudPbkJFA8Zxo7SjDmYbfSxfPXLaIavtkzCccwdNIum+/dRa6KddaUs4Z3Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T01:33:47.152470Z"},"content_sha256":"43a77b9d1367ef6afc899bedf8762525c527ec309860180441e6c7aa4b5aa113","schema_version":"1.0","event_id":"sha256:43a77b9d1367ef6afc899bedf8762525c527ec309860180441e6c7aa4b5aa113"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/bundle.json","state_url":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/65LCIRSIKPVBKBC4AC76JGRQW2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T01:33:47Z","links":{"resolver":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2","bundle":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/bundle.json","state":"https://pith.science/pith/65LCIRSIKPVBKBC4AC76JGRQW2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/65LCIRSIKPVBKBC4AC76JGRQW2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:65LCIRSIKPVBKBC4AC76JGRQW2","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":"4fa63e253fed63c1dcca041ce2f51be1da22a1fe21fb997321c7cf1bd9c7389d","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-11-06T17:36:15Z","title_canon_sha256":"ed938c2b1fa2b89641dfc07c30c46fab10a021a1f2a285d996f630196955d414"},"schema_version":"1.0","source":{"id":"1211.1319","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1319","created_at":"2026-05-18T03:41:21Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1319v1","created_at":"2026-05-18T03:41:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1319","created_at":"2026-05-18T03:41:21Z"},{"alias_kind":"pith_short_12","alias_value":"65LCIRSIKPVB","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"65LCIRSIKPVBKBC4","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"65LCIRSI","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:43a77b9d1367ef6afc899bedf8762525c527ec309860180441e6c7aa4b5aa113","target":"graph","created_at":"2026-05-18T03:41:21Z","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":"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","authors_text":"Laszlo Kozma, Shay Moran","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-11-06T17:36:15Z","title":"Shattering, Graph Orientations, and Connectivity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1319","kind":"arxiv","version":1},"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:b9c5d98d8e30920e59b04553ffe48ff532ba8a9740f94b7b216b46bca36351a9","target":"record","created_at":"2026-05-18T03:41:21Z","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":"4fa63e253fed63c1dcca041ce2f51be1da22a1fe21fb997321c7cf1bd9c7389d","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-11-06T17:36:15Z","title_canon_sha256":"ed938c2b1fa2b89641dfc07c30c46fab10a021a1f2a285d996f630196955d414"},"schema_version":"1.0","source":{"id":"1211.1319","kind":"arxiv","version":1}},"canonical_sha256":"f75624464853ea15045c00bfe49a30b6b3840dcc0fd14ab13167703d885b4ea3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f75624464853ea15045c00bfe49a30b6b3840dcc0fd14ab13167703d885b4ea3","first_computed_at":"2026-05-18T03:41:21.848905Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:21.848905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aC0TyQA/JUIRO2r9IYN4lpq4DOF/yiK1xVX2Hc0dDhavTghjYRPPl/ilfVlwPupav8VHrar3Yq4ukkEtsrkOAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:21.849284Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.1319","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9c5d98d8e30920e59b04553ffe48ff532ba8a9740f94b7b216b46bca36351a9","sha256:43a77b9d1367ef6afc899bedf8762525c527ec309860180441e6c7aa4b5aa113"],"state_sha256":"190d96585b8e5ac8d088e0205c25ec21022f7af4174375b9777299734cc118e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EeyiiKFccC6e9OwyvHI6ue00aXdkO+ftQCDN32Dxwsvvg/EgLqRFnFg7fhJUz0oe3eAajbpIqb++CAo66ryjDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T01:33:47.155248Z","bundle_sha256":"58c6faea712f303d1b557ff2b3e14fdf9537ae0fcb99a6cb214bfae7fb19ec21"}}