{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:YEVENA7EBIEBBONMGQOTSPFSU2","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":"e64fc0ed8c797e6bc2afd01b8a1a855c7e19c23d44c43685bf8b8f3274032e23","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-24T20:06:58Z","title_canon_sha256":"778785619a14fd7239ed2c166e62b300ecd9ced1294e991af276cf0af3a138c4"},"schema_version":"1.0","source":{"id":"1404.6261","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.6261","created_at":"2026-05-18T02:53:17Z"},{"alias_kind":"arxiv_version","alias_value":"1404.6261v1","created_at":"2026-05-18T02:53:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.6261","created_at":"2026-05-18T02:53:17Z"},{"alias_kind":"pith_short_12","alias_value":"YEVENA7EBIEB","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"YEVENA7EBIEBBONM","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"YEVENA7E","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:83bacd9dc7a071aa2fbf8a260ef402f3fb202d69742944bcedbb40f05bd51e0f","target":"graph","created_at":"2026-05-18T02:53:17Z","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":"A {\\em product-injective labeling} of a graph $G$ is an injection $\\chi: V(G) \\to \\mathbb{Z}$ such that $\\chi(u)\\chi(v) \\not= \\chi(x)\\chi(y)$ for any distinct edges $uv, xy\\in E(G)$. Let $P(G)$ be the smallest $N \\geq 1$ such that there exists a product-injective labeling $\\chi : V(G) \\rightarrow [N]$. Let $P(n,d)$ be the maximum possible value of $P(G)$ over $n$-vertex graphs $G$ of maximum degree at most $d$. In this paper, we determine the asymptotic value of $P(n,d)$ for all but a small range of values of $d$ relative to $n$. Specifically, we show that there exist constants $a,b > 0$ such ","authors_text":"Jacques Verstraete, Michael Tait","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-24T20:06:58Z","title":"On sets of integers with restrictions on their products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.6261","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":"sha256:893ee4b825cd6dfdd2609d548e36da32b75cc0c5d8c51398ec22132fc9c3cfed","target":"record","created_at":"2026-05-18T02:53:17Z","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":"e64fc0ed8c797e6bc2afd01b8a1a855c7e19c23d44c43685bf8b8f3274032e23","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-24T20:06:58Z","title_canon_sha256":"778785619a14fd7239ed2c166e62b300ecd9ced1294e991af276cf0af3a138c4"},"schema_version":"1.0","source":{"id":"1404.6261","kind":"arxiv","version":1}},"canonical_sha256":"c12a4683e40a0810b9ac341d393cb2a6b8541bbd939b399acfbbca8a81884645","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c12a4683e40a0810b9ac341d393cb2a6b8541bbd939b399acfbbca8a81884645","first_computed_at":"2026-05-18T02:53:17.029949Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:17.029949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NJGpaxLisvVM5MnggsJaTYFz1VPV2a9g1IWw8c4oAwp2emDlil/5RULCT8kv0jrcxNmI6FnZQbnJImWYMU7GBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:17.030749Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.6261","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:893ee4b825cd6dfdd2609d548e36da32b75cc0c5d8c51398ec22132fc9c3cfed","sha256:83bacd9dc7a071aa2fbf8a260ef402f3fb202d69742944bcedbb40f05bd51e0f"],"state_sha256":"6999a0040d018a974f45c73e2ef0827a37f19c9fb2ca2538321f20bbb4611a79"}