{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:F36V6IC45UERBICHYAPS64C2HF","short_pith_number":"pith:F36V6IC4","canonical_record":{"source":{"id":"1706.01579","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-06T01:54:07Z","cross_cats_sorted":[],"title_canon_sha256":"4e4320b6cecf147645827c87f3804a06be2b92ee3ed553053ea4937eafb69c3b","abstract_canon_sha256":"1af086ce3cb94bd97396b10c0f8bd1f6d2bf0f9b3d41daff68b52d78b9a67ef9"},"schema_version":"1.0"},"canonical_sha256":"2efd5f205ced0910a047c01f2f705a3953facc4603d13410608192f280df96dc","source":{"kind":"arxiv","id":"1706.01579","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.01579","created_at":"2026-05-18T00:42:56Z"},{"alias_kind":"arxiv_version","alias_value":"1706.01579v1","created_at":"2026-05-18T00:42:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.01579","created_at":"2026-05-18T00:42:56Z"},{"alias_kind":"pith_short_12","alias_value":"F36V6IC45UER","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"F36V6IC45UERBICH","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"F36V6IC4","created_at":"2026-05-18T12:31:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:F36V6IC45UERBICHYAPS64C2HF","target":"record","payload":{"canonical_record":{"source":{"id":"1706.01579","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-06T01:54:07Z","cross_cats_sorted":[],"title_canon_sha256":"4e4320b6cecf147645827c87f3804a06be2b92ee3ed553053ea4937eafb69c3b","abstract_canon_sha256":"1af086ce3cb94bd97396b10c0f8bd1f6d2bf0f9b3d41daff68b52d78b9a67ef9"},"schema_version":"1.0"},"canonical_sha256":"2efd5f205ced0910a047c01f2f705a3953facc4603d13410608192f280df96dc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:56.623088Z","signature_b64":"QNaTXiTPON57TMbokPP0GU0HzLyyiG/ZpwYQh0f+J8HhEFUewCYFLmXcTyoNP7Gt60HYFBWFuUGMcLUGGw1kBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2efd5f205ced0910a047c01f2f705a3953facc4603d13410608192f280df96dc","last_reissued_at":"2026-05-18T00:42:56.622376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:56.622376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.01579","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":"2026-05-18T00:42:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6gMh9wb2Lt0HOOQ3r6BaJ22t6CNU/QjCvlBgQC2Lok9tvPTsdhcK+9fHuoAbePPuA9UcQzy7+15UPAyoezMkBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T04:34:42.621267Z"},"content_sha256":"d77f623287e66293c97778fe4115a55eb8a675d252a9dd8f18955484af1bf36e","schema_version":"1.0","event_id":"sha256:d77f623287e66293c97778fe4115a55eb8a675d252a9dd8f18955484af1bf36e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:F36V6IC45UERBICHYAPS64C2HF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Progressions and Paths in Colorings of $\\mathbb Z$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aaron Berger","submitted_at":"2017-06-06T01:54:07Z","abstract_excerpt":"A $\\textit{ladder}$ is a set $S \\subseteq \\mathbb Z^+$ such that any finite coloring of $\\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\\mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $\\textit{accessible}$ and $\\textit{walkable}$ sets, which are sets $S$ such that any coloring of $\\mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01579","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":"2026-05-18T00:42:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"apdC+Y/YT45xV20V6sj47mtazHx4e23hsahpPCB4vBHAfagZJqQA2vvJqmQoY+UTnnuWgjW0OewFT2CORfajDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T04:34:42.621646Z"},"content_sha256":"7ad45111025372b37fd052f58d62c64320c191ea47c55068df4d3931c8b923e0","schema_version":"1.0","event_id":"sha256:7ad45111025372b37fd052f58d62c64320c191ea47c55068df4d3931c8b923e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F36V6IC45UERBICHYAPS64C2HF/bundle.json","state_url":"https://pith.science/pith/F36V6IC45UERBICHYAPS64C2HF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F36V6IC45UERBICHYAPS64C2HF/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":"2026-06-23T04:34:42Z","links":{"resolver":"https://pith.science/pith/F36V6IC45UERBICHYAPS64C2HF","bundle":"https://pith.science/pith/F36V6IC45UERBICHYAPS64C2HF/bundle.json","state":"https://pith.science/pith/F36V6IC45UERBICHYAPS64C2HF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F36V6IC45UERBICHYAPS64C2HF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:F36V6IC45UERBICHYAPS64C2HF","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":"1af086ce3cb94bd97396b10c0f8bd1f6d2bf0f9b3d41daff68b52d78b9a67ef9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-06T01:54:07Z","title_canon_sha256":"4e4320b6cecf147645827c87f3804a06be2b92ee3ed553053ea4937eafb69c3b"},"schema_version":"1.0","source":{"id":"1706.01579","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.01579","created_at":"2026-05-18T00:42:56Z"},{"alias_kind":"arxiv_version","alias_value":"1706.01579v1","created_at":"2026-05-18T00:42:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.01579","created_at":"2026-05-18T00:42:56Z"},{"alias_kind":"pith_short_12","alias_value":"F36V6IC45UER","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"F36V6IC45UERBICH","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"F36V6IC4","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:7ad45111025372b37fd052f58d62c64320c191ea47c55068df4d3931c8b923e0","target":"graph","created_at":"2026-05-18T00:42:56Z","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":"A $\\textit{ladder}$ is a set $S \\subseteq \\mathbb Z^+$ such that any finite coloring of $\\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\\mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $\\textit{accessible}$ and $\\textit{walkable}$ sets, which are sets $S$ such that any coloring of $\\mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 a","authors_text":"Aaron Berger","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-06T01:54:07Z","title":"Progressions and Paths in Colorings of $\\mathbb Z$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01579","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:d77f623287e66293c97778fe4115a55eb8a675d252a9dd8f18955484af1bf36e","target":"record","created_at":"2026-05-18T00:42:56Z","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":"1af086ce3cb94bd97396b10c0f8bd1f6d2bf0f9b3d41daff68b52d78b9a67ef9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-06T01:54:07Z","title_canon_sha256":"4e4320b6cecf147645827c87f3804a06be2b92ee3ed553053ea4937eafb69c3b"},"schema_version":"1.0","source":{"id":"1706.01579","kind":"arxiv","version":1}},"canonical_sha256":"2efd5f205ced0910a047c01f2f705a3953facc4603d13410608192f280df96dc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2efd5f205ced0910a047c01f2f705a3953facc4603d13410608192f280df96dc","first_computed_at":"2026-05-18T00:42:56.622376Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:56.622376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QNaTXiTPON57TMbokPP0GU0HzLyyiG/ZpwYQh0f+J8HhEFUewCYFLmXcTyoNP7Gt60HYFBWFuUGMcLUGGw1kBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:56.623088Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.01579","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d77f623287e66293c97778fe4115a55eb8a675d252a9dd8f18955484af1bf36e","sha256:7ad45111025372b37fd052f58d62c64320c191ea47c55068df4d3931c8b923e0"],"state_sha256":"3a5c7b140b9589bac14fb0f11395be4f09109a7a99d280e00f37799acefd6645"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x7BA4jC2NE1Pi8GPENJgSyBQaq9ZDfeHDPVcwpQns6Z81eZXYgdWqTQv/y0JUmMd+I7UqsLPykshgAFGY+vtDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T04:34:42.623736Z","bundle_sha256":"cb60755d462be5ed9540f7af36406863d18a71069b7413bc6a26bc1722539189"}}