{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:S4I4UUXYM7SEGBSVQH7ZU4QAV4","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":"80340065aa5febb525606e4a05788b44f16fe382d3f362f3e68bfc8c6f56db7b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-12T18:52:39Z","title_canon_sha256":"3eaaf74209c4a4a8bd32d8bcc16bbdb9819b2c68081441357dfaca957ddd5118"},"schema_version":"1.0","source":{"id":"1807.04785","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.04785","created_at":"2026-05-18T00:10:51Z"},{"alias_kind":"arxiv_version","alias_value":"1807.04785v1","created_at":"2026-05-18T00:10:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.04785","created_at":"2026-05-18T00:10:51Z"},{"alias_kind":"pith_short_12","alias_value":"S4I4UUXYM7SE","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"S4I4UUXYM7SEGBSV","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"S4I4UUXY","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:54c2e26c11bc5d3fae0308ef230bd21dbfe084a16b69d18e857e5482c79b20ae","target":"graph","created_at":"2026-05-18T00:10:51Z","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":"The Harborth constant of a finite group $G$ is the smallest integer $k\\geq \\exp(G)$ such that any subset of $G$ of size $k$ contains $\\exp(G)$ distinct elements whose product is $1$. Generalizing previous work on the Harborth constants of dihedral groups, we compute the Harborth constants for the metacyclic groups of the form $H_{n, m}=\\langle x, y \\mid x^n=1, y^2=x^m, yx=x^{-1}y \\rangle$. We also solve the \"inverse\" problem of characterizing all smaller subsets that do not contain $\\exp(H_{n,m})$ distinct elements whose product is $1$.","authors_text":"Noah Kravitz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-12T18:52:39Z","title":"Harborth Constants for Certain Classes of Metacyclic Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04785","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":"sha256:1bc3cd1b42ebe81ae1c479e174a0e471be4c796a0bfb15c300cd088472ae2b75","target":"record","created_at":"2026-05-18T00:10:51Z","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":"80340065aa5febb525606e4a05788b44f16fe382d3f362f3e68bfc8c6f56db7b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-12T18:52:39Z","title_canon_sha256":"3eaaf74209c4a4a8bd32d8bcc16bbdb9819b2c68081441357dfaca957ddd5118"},"schema_version":"1.0","source":{"id":"1807.04785","kind":"arxiv","version":1}},"canonical_sha256":"9711ca52f867e443065581ff9a7200af1a3e02ff4a0825dc1746414c09c60d12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9711ca52f867e443065581ff9a7200af1a3e02ff4a0825dc1746414c09c60d12","first_computed_at":"2026-05-18T00:10:51.440926Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:51.440926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QeBrsviXwtqiT+nvH2DVOTM0MMcVF9Xin2OzD03z9Ot44VHnnLJlfdS+HMXhiFOvY8BoyIJfUlT4xo4wl4tJCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:51.441443Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.04785","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1bc3cd1b42ebe81ae1c479e174a0e471be4c796a0bfb15c300cd088472ae2b75","sha256:54c2e26c11bc5d3fae0308ef230bd21dbfe084a16b69d18e857e5482c79b20ae"],"state_sha256":"066f5f1338cf4946139c8785408712e54391fe40ef03baca290f2c5f8069505a"}