{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:Y5TMX7RAK6W2YVW4TH2ICVDR7F","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":"fd2b464a59da2a0d40bbde08935052f694d4b1643eb1bbf38ccb77997ae875cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-16T14:39:07Z","title_canon_sha256":"d803cf32be5b9e129b728b8ca9a06dfcfbde16b8ff8c3420eb6b4476cc8daec2"},"schema_version":"1.0","source":{"id":"1705.05730","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.05730","created_at":"2026-05-18T00:44:20Z"},{"alias_kind":"arxiv_version","alias_value":"1705.05730v1","created_at":"2026-05-18T00:44:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.05730","created_at":"2026-05-18T00:44:20Z"},{"alias_kind":"pith_short_12","alias_value":"Y5TMX7RAK6W2","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"Y5TMX7RAK6W2YVW4","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"Y5TMX7RA","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:35f9da7fb0e41355554adeb00effbe3b22b0913b8193f8a18b233315ccdaf801","target":"graph","created_at":"2026-05-18T00:44:20Z","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":"Let $\\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\\cap \\{1,2,\\ldots,n\\}$. Let $f(n, k) = \\text{max}_{A \\in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \\dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erd\\H{o}s conjectured that $f(n, k) = |E(n,k)|$ for every $n \\ge p_{k}$. Recently Chen a","authors_text":"Csaba S\\'andor, Quan-Hui Yang, S\\'andor Z. Kiss","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-16T14:39:07Z","title":"On a conjecture of Erd\\H{o}s about sets without $k$ pairwise coprime integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.05730","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:c23199052b07b5d184c4d779059d200f441ebe9d803eca7d5fe84a702b3acc50","target":"record","created_at":"2026-05-18T00:44:20Z","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":"fd2b464a59da2a0d40bbde08935052f694d4b1643eb1bbf38ccb77997ae875cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-16T14:39:07Z","title_canon_sha256":"d803cf32be5b9e129b728b8ca9a06dfcfbde16b8ff8c3420eb6b4476cc8daec2"},"schema_version":"1.0","source":{"id":"1705.05730","kind":"arxiv","version":1}},"canonical_sha256":"c766cbfe2057adac56dc99f4815471f95cf96f2f0919719cf8b516fbd707693a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c766cbfe2057adac56dc99f4815471f95cf96f2f0919719cf8b516fbd707693a","first_computed_at":"2026-05-18T00:44:20.338521Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:20.338521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cuJn0Ladjf1J7C7tcViid6viUpe2X89NpU2Lx/jKrb6YOqJcAithV+Ka2k/HY8HAgIY7bpFhxN1J4LLjb0nRDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:20.339022Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.05730","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c23199052b07b5d184c4d779059d200f441ebe9d803eca7d5fe84a702b3acc50","sha256:35f9da7fb0e41355554adeb00effbe3b22b0913b8193f8a18b233315ccdaf801"],"state_sha256":"f4507f542c774da7ecc003837f7b558cad29b156b308a617cad5db9ab63d21d7"}