{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:76K4CZRU6D6CTIRTDJARXBELUZ","short_pith_number":"pith:76K4CZRU","canonical_record":{"source":{"id":"1902.08259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-21T20:54:28Z","cross_cats_sorted":[],"title_canon_sha256":"9d16aa07247893f7b92c6ed3cc2d7a747b3aa14ffbebf66d05f29f8a59ac2bc2","abstract_canon_sha256":"6025bcb29f50552ccc2064c050b7e74554487e782fd474ed1213947ff06787f6"},"schema_version":"1.0"},"canonical_sha256":"ff95c16634f0fc29a2331a411b848ba66f9a7957021e320aac2cb9418c9ad4bb","source":{"kind":"arxiv","id":"1902.08259","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.08259","created_at":"2026-05-17T23:52:57Z"},{"alias_kind":"arxiv_version","alias_value":"1902.08259v1","created_at":"2026-05-17T23:52:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.08259","created_at":"2026-05-17T23:52:57Z"},{"alias_kind":"pith_short_12","alias_value":"76K4CZRU6D6C","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"76K4CZRU6D6CTIRT","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"76K4CZRU","created_at":"2026-05-18T12:33:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:76K4CZRU6D6CTIRTDJARXBELUZ","target":"record","payload":{"canonical_record":{"source":{"id":"1902.08259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-21T20:54:28Z","cross_cats_sorted":[],"title_canon_sha256":"9d16aa07247893f7b92c6ed3cc2d7a747b3aa14ffbebf66d05f29f8a59ac2bc2","abstract_canon_sha256":"6025bcb29f50552ccc2064c050b7e74554487e782fd474ed1213947ff06787f6"},"schema_version":"1.0"},"canonical_sha256":"ff95c16634f0fc29a2331a411b848ba66f9a7957021e320aac2cb9418c9ad4bb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:57.843854Z","signature_b64":"aor35x3FSMhMBzhBclotOnwXwdYr9YEqTwkjpSt9y3RYBa8W+7r4ViwXPI7xI/r9/JhypRKu8gNPpnc2UQHiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff95c16634f0fc29a2331a411b848ba66f9a7957021e320aac2cb9418c9ad4bb","last_reissued_at":"2026-05-17T23:52:57.843158Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:57.843158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1902.08259","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":"2026-05-17T23:52:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rAFqyC97v09Q/KiP+Lv/aOJWdF+58d+7GJuEk1u0KmEtIWDp3dmoJgOoScU4PTR0NXtL0z7OAEQ92sZQmYGnAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:37:08.507452Z"},"content_sha256":"786de342bfc16fb3f2c1b474dbdbec2451abda9c65014d18f280b1680f8200c1","schema_version":"1.0","event_id":"sha256:786de342bfc16fb3f2c1b474dbdbec2451abda9c65014d18f280b1680f8200c1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:76K4CZRU6D6CTIRTDJARXBELUZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Number of Discrete Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Eyvindur Ari Palsson, Steven Senger","submitted_at":"2019-02-21T20:54:28Z","abstract_excerpt":"We study a generalization of Erd\\H os's unit distances problem to chains of $k$ distances. Given $\\mathcal P,$ a set of $n$ points, and a sequence of distances $(\\delta_1,\\ldots,\\delta_k)$, we study the maximum possible number of tuples of distinct points $(p_1,\\ldots,p_{k+1})\\in \\mathcal P^{k+1}$ satisfying $|p_j p_{j+1}|=\\delta_j$ for every $1\\leq j \\leq k$. We study the problem in $\\mathbb R^2$ and in $\\mathbb R^3$, and derive upper and lower bounds for this family of problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.08259","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":"2026-05-17T23:52:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xHOEVOuhFcPPh+x1K3tp0NHYX82xnXxeckjf27c9Ef4mXzgWagBwdBSZ0RnL4wC9Mt2oCXa03XsIMseVA5HDCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:37:08.507803Z"},"content_sha256":"309b27baef9129e7b189efeefb54510399bbfbb24d6b31fd4a2a9e4e9929474f","schema_version":"1.0","event_id":"sha256:309b27baef9129e7b189efeefb54510399bbfbb24d6b31fd4a2a9e4e9929474f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/76K4CZRU6D6CTIRTDJARXBELUZ/bundle.json","state_url":"https://pith.science/pith/76K4CZRU6D6CTIRTDJARXBELUZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/76K4CZRU6D6CTIRTDJARXBELUZ/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":"2026-06-03T06:37:08Z","links":{"resolver":"https://pith.science/pith/76K4CZRU6D6CTIRTDJARXBELUZ","bundle":"https://pith.science/pith/76K4CZRU6D6CTIRTDJARXBELUZ/bundle.json","state":"https://pith.science/pith/76K4CZRU6D6CTIRTDJARXBELUZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/76K4CZRU6D6CTIRTDJARXBELUZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:76K4CZRU6D6CTIRTDJARXBELUZ","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":"6025bcb29f50552ccc2064c050b7e74554487e782fd474ed1213947ff06787f6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-21T20:54:28Z","title_canon_sha256":"9d16aa07247893f7b92c6ed3cc2d7a747b3aa14ffbebf66d05f29f8a59ac2bc2"},"schema_version":"1.0","source":{"id":"1902.08259","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.08259","created_at":"2026-05-17T23:52:57Z"},{"alias_kind":"arxiv_version","alias_value":"1902.08259v1","created_at":"2026-05-17T23:52:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.08259","created_at":"2026-05-17T23:52:57Z"},{"alias_kind":"pith_short_12","alias_value":"76K4CZRU6D6C","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"76K4CZRU6D6CTIRT","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"76K4CZRU","created_at":"2026-05-18T12:33:12Z"}],"graph_snapshots":[{"event_id":"sha256:309b27baef9129e7b189efeefb54510399bbfbb24d6b31fd4a2a9e4e9929474f","target":"graph","created_at":"2026-05-17T23:52:57Z","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":"We study a generalization of Erd\\H os's unit distances problem to chains of $k$ distances. Given $\\mathcal P,$ a set of $n$ points, and a sequence of distances $(\\delta_1,\\ldots,\\delta_k)$, we study the maximum possible number of tuples of distinct points $(p_1,\\ldots,p_{k+1})\\in \\mathcal P^{k+1}$ satisfying $|p_j p_{j+1}|=\\delta_j$ for every $1\\leq j \\leq k$. We study the problem in $\\mathbb R^2$ and in $\\mathbb R^3$, and derive upper and lower bounds for this family of problems.","authors_text":"Adam Sheffer, Eyvindur Ari Palsson, Steven Senger","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-21T20:54:28Z","title":"On the Number of Discrete Chains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.08259","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":"sha256:786de342bfc16fb3f2c1b474dbdbec2451abda9c65014d18f280b1680f8200c1","target":"record","created_at":"2026-05-17T23:52:57Z","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":"6025bcb29f50552ccc2064c050b7e74554487e782fd474ed1213947ff06787f6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-21T20:54:28Z","title_canon_sha256":"9d16aa07247893f7b92c6ed3cc2d7a747b3aa14ffbebf66d05f29f8a59ac2bc2"},"schema_version":"1.0","source":{"id":"1902.08259","kind":"arxiv","version":1}},"canonical_sha256":"ff95c16634f0fc29a2331a411b848ba66f9a7957021e320aac2cb9418c9ad4bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff95c16634f0fc29a2331a411b848ba66f9a7957021e320aac2cb9418c9ad4bb","first_computed_at":"2026-05-17T23:52:57.843158Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:57.843158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aor35x3FSMhMBzhBclotOnwXwdYr9YEqTwkjpSt9y3RYBa8W+7r4ViwXPI7xI/r9/JhypRKu8gNPpnc2UQHiAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:57.843854Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.08259","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:786de342bfc16fb3f2c1b474dbdbec2451abda9c65014d18f280b1680f8200c1","sha256:309b27baef9129e7b189efeefb54510399bbfbb24d6b31fd4a2a9e4e9929474f"],"state_sha256":"d2f29945ecc75a5778764b1a7e26a372ad271b21fdee9f9047c63f2205e847bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mTVYQVft5t32zTTdHE1b/n81gXhI0PjWAqSt2Gu9EwwxSl2bFHRl5r+nHAaAVWtLfEPvvEzGk99fYD/ZD/58CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T06:37:08.509644Z","bundle_sha256":"df2e950fe733c3e9029146f3603213b65e4c62853f8077e3623286d16088a165"}}