{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:QOHDT5RSQ4JWPSGCE4Z4FIBP26","short_pith_number":"pith:QOHDT5RS","canonical_record":{"source":{"id":"1803.00861","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-02T14:23:44Z","cross_cats_sorted":[],"title_canon_sha256":"4c238b6ecd691a127f951ebddd519c7fd7c815b16bbb34cf308662fdfadf666a","abstract_canon_sha256":"1089ffe939173d609d17cae22efc7c6a05fc8d0429b8379fe5187b954244e53f"},"schema_version":"1.0"},"canonical_sha256":"838e39f632871367c8c22733c2a02fd7bf93b9e7df0c0035e688b0f50ee1a361","source":{"kind":"arxiv","id":"1803.00861","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.00861","created_at":"2026-05-18T00:21:45Z"},{"alias_kind":"arxiv_version","alias_value":"1803.00861v2","created_at":"2026-05-18T00:21:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.00861","created_at":"2026-05-18T00:21:45Z"},{"alias_kind":"pith_short_12","alias_value":"QOHDT5RSQ4JW","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QOHDT5RSQ4JWPSGC","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QOHDT5RS","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:QOHDT5RSQ4JWPSGCE4Z4FIBP26","target":"record","payload":{"canonical_record":{"source":{"id":"1803.00861","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-02T14:23:44Z","cross_cats_sorted":[],"title_canon_sha256":"4c238b6ecd691a127f951ebddd519c7fd7c815b16bbb34cf308662fdfadf666a","abstract_canon_sha256":"1089ffe939173d609d17cae22efc7c6a05fc8d0429b8379fe5187b954244e53f"},"schema_version":"1.0"},"canonical_sha256":"838e39f632871367c8c22733c2a02fd7bf93b9e7df0c0035e688b0f50ee1a361","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:45.502501Z","signature_b64":"WohlzVNrHNxekUO1wsIZcw6yR4Kq4+3zH5kKpkKqMV/3nhHeMY460nU1nmcRMPL0gxqBHGHbCdgH6bnXnHDADA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"838e39f632871367c8c22733c2a02fd7bf93b9e7df0c0035e688b0f50ee1a361","last_reissued_at":"2026-05-18T00:21:45.501911Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:45.501911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.00861","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:21:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fNHDdOj3iAayn5MWv3cu0NzbC2FdnVMg9URZLiD196SPHwfAkKY+SeSH58nMKJQUUw0glIccSiVLBD1owuZcDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:53:25.107674Z"},"content_sha256":"5beef563fa8aadc477ae0e6ce45f82592a6b398c086f4b5dcd9e6f8b5d4a0b2e","schema_version":"1.0","event_id":"sha256:5beef563fa8aadc477ae0e6ce45f82592a6b398c086f4b5dcd9e6f8b5d4a0b2e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:QOHDT5RSQ4JWPSGCE4Z4FIBP26","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Determination of 2-color zero-sum generalized Schur Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aaron Robertson, Bidisha Roy, Subha Sarkar","submitted_at":"2018-03-02T14:23:44Z","abstract_excerpt":"Consider the equation $\\mathcal{E}: x_1+ \\cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $r\\mid k$. The number $S_{\\mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such that for any 2-coloring $\\chi: [1, t] \\to \\{0, 1\\}$ there exists a solution $(\\hat{x}_1, \\hat{x}_2, \\ldots, \\hat{x}_k)$ to the equation $\\mathcal{E}$ satisfying $\\displaystyle \\sum_{i=1}^k\\chi(\\hat{x}_i) \\equiv 0\\pmod{r}$. In a recent paper, the first author posed the question of determining the exact value of $S_{\\mathfrak{z}, 2}(k;4)$. In this article, we solve this problem and s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.00861","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:21:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qpXQuAbNLijxCmsDkvWT3uijpJqHB1XDO6Evp69myTTqQ7sx+02HmTVyZZA/NvCqhfzx9ajhWbKFAwmoUthyDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:53:25.108507Z"},"content_sha256":"36a4308fd9a255ffe6e815431a686952184ee6d51e61191e490d9c186b50ccfd","schema_version":"1.0","event_id":"sha256:36a4308fd9a255ffe6e815431a686952184ee6d51e61191e490d9c186b50ccfd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QOHDT5RSQ4JWPSGCE4Z4FIBP26/bundle.json","state_url":"https://pith.science/pith/QOHDT5RSQ4JWPSGCE4Z4FIBP26/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QOHDT5RSQ4JWPSGCE4Z4FIBP26/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":"2026-06-11T07:53:25Z","links":{"resolver":"https://pith.science/pith/QOHDT5RSQ4JWPSGCE4Z4FIBP26","bundle":"https://pith.science/pith/QOHDT5RSQ4JWPSGCE4Z4FIBP26/bundle.json","state":"https://pith.science/pith/QOHDT5RSQ4JWPSGCE4Z4FIBP26/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QOHDT5RSQ4JWPSGCE4Z4FIBP26/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QOHDT5RSQ4JWPSGCE4Z4FIBP26","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":"1089ffe939173d609d17cae22efc7c6a05fc8d0429b8379fe5187b954244e53f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-02T14:23:44Z","title_canon_sha256":"4c238b6ecd691a127f951ebddd519c7fd7c815b16bbb34cf308662fdfadf666a"},"schema_version":"1.0","source":{"id":"1803.00861","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.00861","created_at":"2026-05-18T00:21:45Z"},{"alias_kind":"arxiv_version","alias_value":"1803.00861v2","created_at":"2026-05-18T00:21:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.00861","created_at":"2026-05-18T00:21:45Z"},{"alias_kind":"pith_short_12","alias_value":"QOHDT5RSQ4JW","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QOHDT5RSQ4JWPSGC","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QOHDT5RS","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:36a4308fd9a255ffe6e815431a686952184ee6d51e61191e490d9c186b50ccfd","target":"graph","created_at":"2026-05-18T00:21:45Z","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":"Consider the equation $\\mathcal{E}: x_1+ \\cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $r\\mid k$. The number $S_{\\mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such that for any 2-coloring $\\chi: [1, t] \\to \\{0, 1\\}$ there exists a solution $(\\hat{x}_1, \\hat{x}_2, \\ldots, \\hat{x}_k)$ to the equation $\\mathcal{E}$ satisfying $\\displaystyle \\sum_{i=1}^k\\chi(\\hat{x}_i) \\equiv 0\\pmod{r}$. In a recent paper, the first author posed the question of determining the exact value of $S_{\\mathfrak{z}, 2}(k;4)$. In this article, we solve this problem and s","authors_text":"Aaron Robertson, Bidisha Roy, Subha Sarkar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-02T14:23:44Z","title":"The Determination of 2-color zero-sum generalized Schur Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.00861","kind":"arxiv","version":2},"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":"sha256:5beef563fa8aadc477ae0e6ce45f82592a6b398c086f4b5dcd9e6f8b5d4a0b2e","target":"record","created_at":"2026-05-18T00:21:45Z","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":"1089ffe939173d609d17cae22efc7c6a05fc8d0429b8379fe5187b954244e53f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-02T14:23:44Z","title_canon_sha256":"4c238b6ecd691a127f951ebddd519c7fd7c815b16bbb34cf308662fdfadf666a"},"schema_version":"1.0","source":{"id":"1803.00861","kind":"arxiv","version":2}},"canonical_sha256":"838e39f632871367c8c22733c2a02fd7bf93b9e7df0c0035e688b0f50ee1a361","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"838e39f632871367c8c22733c2a02fd7bf93b9e7df0c0035e688b0f50ee1a361","first_computed_at":"2026-05-18T00:21:45.501911Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:45.501911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WohlzVNrHNxekUO1wsIZcw6yR4Kq4+3zH5kKpkKqMV/3nhHeMY460nU1nmcRMPL0gxqBHGHbCdgH6bnXnHDADA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:45.502501Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.00861","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5beef563fa8aadc477ae0e6ce45f82592a6b398c086f4b5dcd9e6f8b5d4a0b2e","sha256:36a4308fd9a255ffe6e815431a686952184ee6d51e61191e490d9c186b50ccfd"],"state_sha256":"be628eea200c5327660fb0acf19bc9ce11eb135064074bc49c13f15f64026f56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Cc+1zyMCdzjPbTtRbENcbuSizTEq1kBF26eR0TLckZWS1zYPMHuke8Z2EKx+aRX0oZq2WLYibwRNG+hJkXpAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T07:53:25.112475Z","bundle_sha256":"6c9eaa9d6aaf5343a485d2b1f1bdf9ae8d0f9ead9d47278512e080897a83a61f"}}