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Comput., 1994) proved that if $G_1(\\mathcal S)=(V,E)$ is the subgraph of the hypercube $Q_m$ induced by $\\mathcal S$ (called the 1-inclusion graph of $\\mathcal S$), then $\\frac{|E|}{|V|}\\le \\text{VC-dim}({\\mathcal S})$. Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation.\n  In this note, we adapt the shifting technique to prove that if $\\math"},"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":"1711.11414","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-11-30T14:21:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"048de76cd28b29733de41cc8f621bd606abb47073e8ed33b551f23b5fd938af6","abstract_canon_sha256":"26610d26cfeb8991a81631488241a624f70990c19a47d31f18e69eca7d3a0d32"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:40.120621Z","signature_b64":"RAGltATnymAM5Lv02Z6n/Ii9NAGhCC4NZUSfPumEZEEXgA41TAoJMPaU6m8sPa+qyPCPy4ONutA6DAvIF4psBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b71ad1d2a57bf16bc0b940d3792552de94ffdb6c2dd6b7cfd6542d0c86ae8bec","last_reissued_at":"2026-05-18T00:28:40.119995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:40.119995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On density of subgraphs of halved cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Arnaud Labourel, S\\'ebastien Ratel, Victor Chepoi","submitted_at":"2017-11-30T14:21:59Z","abstract_excerpt":"Let $\\mathcal S$ be a family of subsets of a set $X$ of cardinality $m$ and $\\text{VC-dim}(\\mathcal S)$ be the Vapnik-Chervonenkis dimension of $\\mathcal S$. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if $G_1(\\mathcal S)=(V,E)$ is the subgraph of the hypercube $Q_m$ induced by $\\mathcal S$ (called the 1-inclusion graph of $\\mathcal S$), then $\\frac{|E|}{|V|}\\le \\text{VC-dim}({\\mathcal S})$. Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation.\n  In this note, we adapt the shifting technique to prove that if $\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11414","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1711.11414","created_at":"2026-05-18T00:28:40.120083+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.11414v2","created_at":"2026-05-18T00:28:40.120083+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.11414","created_at":"2026-05-18T00:28:40.120083+00:00"},{"alias_kind":"pith_short_12","alias_value":"W4NNDUVFPPYW","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"W4NNDUVFPPYWXQFZ","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"W4NNDUVF","created_at":"2026-05-18T12:31:49.984773+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/W4NNDUVFPPYWXQFZIDJXSJKS32","json":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32.json","graph_json":"https://pith.science/api/pith-number/W4NNDUVFPPYWXQFZIDJXSJKS32/graph.json","events_json":"https://pith.science/api/pith-number/W4NNDUVFPPYWXQFZIDJXSJKS32/events.json","paper":"https://pith.science/paper/W4NNDUVF"},"agent_actions":{"view_html":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32","download_json":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32.json","view_paper":"https://pith.science/paper/W4NNDUVF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.11414&json=true","fetch_graph":"https://pith.science/api/pith-number/W4NNDUVFPPYWXQFZIDJXSJKS32/graph.json","fetch_events":"https://pith.science/api/pith-number/W4NNDUVFPPYWXQFZIDJXSJKS32/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32/action/storage_attestation","attest_author":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32/action/author_attestation","sign_citation":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32/action/citation_signature","submit_replication":"https://pith.science/pith/W4NNDUVFPPYWXQFZIDJXSJKS32/action/replication_record"}},"created_at":"2026-05-18T00:28:40.120083+00:00","updated_at":"2026-05-18T00:28:40.120083+00:00"}