{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:GHYYUWRVM7NNLG3NSH5HHAZSZA","short_pith_number":"pith:GHYYUWRV","canonical_record":{"source":{"id":"1710.00083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-29T20:34:02Z","cross_cats_sorted":[],"title_canon_sha256":"a443183004bc405777dfcf6aaf19f4e5b21627b16521f290d82e91428b811509","abstract_canon_sha256":"f751aefd605eceb225f4f5a0ed9ace9671a9c44a49a283f62f1f404bc489dad4"},"schema_version":"1.0"},"canonical_sha256":"31f18a5a3567dad59b6d91fa738332c80d29984627fa2c5efeff9ba94dc52083","source":{"kind":"arxiv","id":"1710.00083","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.00083","created_at":"2026-05-18T00:33:57Z"},{"alias_kind":"arxiv_version","alias_value":"1710.00083v1","created_at":"2026-05-18T00:33:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.00083","created_at":"2026-05-18T00:33:57Z"},{"alias_kind":"pith_short_12","alias_value":"GHYYUWRVM7NN","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"GHYYUWRVM7NNLG3N","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"GHYYUWRV","created_at":"2026-05-18T12:31:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:GHYYUWRVM7NNLG3NSH5HHAZSZA","target":"record","payload":{"canonical_record":{"source":{"id":"1710.00083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-29T20:34:02Z","cross_cats_sorted":[],"title_canon_sha256":"a443183004bc405777dfcf6aaf19f4e5b21627b16521f290d82e91428b811509","abstract_canon_sha256":"f751aefd605eceb225f4f5a0ed9ace9671a9c44a49a283f62f1f404bc489dad4"},"schema_version":"1.0"},"canonical_sha256":"31f18a5a3567dad59b6d91fa738332c80d29984627fa2c5efeff9ba94dc52083","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:57.968931Z","signature_b64":"Rg+ntB3XhpRyG2lefKSSkLYsM/fkv6ukJUfo7tUr3uhiO2t8FMH02DXSgCj8ENNW7S5WXiYq8Zqnm95MH5a1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31f18a5a3567dad59b6d91fa738332c80d29984627fa2c5efeff9ba94dc52083","last_reissued_at":"2026-05-18T00:33:57.968289Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:57.968289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.00083","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":"2026-05-18T00:33:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tDeEbdA58VNMNVBot+dmNPvPKRsdwBhf49KkbQh9y7yJ4xJQzDxwn9aaWmgXZcE+fbjttxqDu5GfxhyResaZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:15:59.408744Z"},"content_sha256":"74140f20a7309dfbf11798c5cd7c31eaf31ede408c369185028e0f33be2f129a","schema_version":"1.0","event_id":"sha256:74140f20a7309dfbf11798c5cd7c31eaf31ede408c369185028e0f33be2f129a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:GHYYUWRVM7NNLG3NSH5HHAZSZA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extremal Threshold Graphs for Matchings and Independent Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A.J. Radcliffe, L. Keough","submitted_at":"2017-09-29T20:34:02Z","abstract_excerpt":"Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed $k\\ge 1$, among all graphs on $n$ vertices with $m$ edges, some threshold graph has the fewest matchings of size $k$; indeed either the lex graph or the colex graph is such an extremal example. In this paper we consider the problem of maximizing the number of matchings in the class of threshold graphs. We prove that the minimizers are what we call \\emph{almost alternating threshold graphs}.\n  We also discuss a problem with a similar flavor: which threshold "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.00083","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":"2026-05-18T00:33:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Iew8ZS4+u6tV/+9rhnRrPqtyvawT/hXSmenrhyLwhN5a7Jpmwv07ICIVDqvUiYvbsVPd6MY8zazkhx+cCttXBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:15:59.409093Z"},"content_sha256":"a698ae3f769fc8fa3b60605a03950b2a1603eb63a0ee398ecad52502a3ab588a","schema_version":"1.0","event_id":"sha256:a698ae3f769fc8fa3b60605a03950b2a1603eb63a0ee398ecad52502a3ab588a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GHYYUWRVM7NNLG3NSH5HHAZSZA/bundle.json","state_url":"https://pith.science/pith/GHYYUWRVM7NNLG3NSH5HHAZSZA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GHYYUWRVM7NNLG3NSH5HHAZSZA/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":"2026-06-02T16:15:59Z","links":{"resolver":"https://pith.science/pith/GHYYUWRVM7NNLG3NSH5HHAZSZA","bundle":"https://pith.science/pith/GHYYUWRVM7NNLG3NSH5HHAZSZA/bundle.json","state":"https://pith.science/pith/GHYYUWRVM7NNLG3NSH5HHAZSZA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GHYYUWRVM7NNLG3NSH5HHAZSZA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:GHYYUWRVM7NNLG3NSH5HHAZSZA","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":"f751aefd605eceb225f4f5a0ed9ace9671a9c44a49a283f62f1f404bc489dad4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-29T20:34:02Z","title_canon_sha256":"a443183004bc405777dfcf6aaf19f4e5b21627b16521f290d82e91428b811509"},"schema_version":"1.0","source":{"id":"1710.00083","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.00083","created_at":"2026-05-18T00:33:57Z"},{"alias_kind":"arxiv_version","alias_value":"1710.00083v1","created_at":"2026-05-18T00:33:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.00083","created_at":"2026-05-18T00:33:57Z"},{"alias_kind":"pith_short_12","alias_value":"GHYYUWRVM7NN","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"GHYYUWRVM7NNLG3N","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"GHYYUWRV","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:a698ae3f769fc8fa3b60605a03950b2a1603eb63a0ee398ecad52502a3ab588a","target":"graph","created_at":"2026-05-18T00:33:57Z","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":"Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed $k\\ge 1$, among all graphs on $n$ vertices with $m$ edges, some threshold graph has the fewest matchings of size $k$; indeed either the lex graph or the colex graph is such an extremal example. In this paper we consider the problem of maximizing the number of matchings in the class of threshold graphs. We prove that the minimizers are what we call \\emph{almost alternating threshold graphs}.\n  We also discuss a problem with a similar flavor: which threshold ","authors_text":"A.J. Radcliffe, L. Keough","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-29T20:34:02Z","title":"Extremal Threshold Graphs for Matchings and Independent Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.00083","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":"sha256:74140f20a7309dfbf11798c5cd7c31eaf31ede408c369185028e0f33be2f129a","target":"record","created_at":"2026-05-18T00:33:57Z","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":"f751aefd605eceb225f4f5a0ed9ace9671a9c44a49a283f62f1f404bc489dad4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-29T20:34:02Z","title_canon_sha256":"a443183004bc405777dfcf6aaf19f4e5b21627b16521f290d82e91428b811509"},"schema_version":"1.0","source":{"id":"1710.00083","kind":"arxiv","version":1}},"canonical_sha256":"31f18a5a3567dad59b6d91fa738332c80d29984627fa2c5efeff9ba94dc52083","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31f18a5a3567dad59b6d91fa738332c80d29984627fa2c5efeff9ba94dc52083","first_computed_at":"2026-05-18T00:33:57.968289Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:57.968289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rg+ntB3XhpRyG2lefKSSkLYsM/fkv6ukJUfo7tUr3uhiO2t8FMH02DXSgCj8ENNW7S5WXiYq8Zqnm95MH5a1BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:57.968931Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.00083","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74140f20a7309dfbf11798c5cd7c31eaf31ede408c369185028e0f33be2f129a","sha256:a698ae3f769fc8fa3b60605a03950b2a1603eb63a0ee398ecad52502a3ab588a"],"state_sha256":"f5148fffaf016160d0bc73182e0d619d47ddf1bee1aecc2ec3e30e0f750fadf5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iB9NuhZcUOAfzBvkSt2BTM4ThF5C+M9UOC2b7SCWQUSsSaLTM7Asoc2LkvSr53OVya6jjGQsal0Ex1sXM4lcBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T16:15:59.411041Z","bundle_sha256":"e76070ef612d32cd4b7a340d5e27b5f0f969769e70cdb08627fdacd85e2377d8"}}