{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1993:UUVC4HX5QHV5S3Q2V76LL247PP","short_pith_number":"pith:UUVC4HX5","canonical_record":{"source":{"id":"math/9301209","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"ea37261527356fa26ecc2e637470b6b32456eb8c18312930c29bb4e9464b68e6","abstract_canon_sha256":"047fe888b383cc31616a2fb208a2cfc62e4c146fd5c38ba18e2b70878c12064b"},"schema_version":"1.0"},"canonical_sha256":"a52a2e1efd81ebd96e1aaffcb5eb9f7beca96fbf60cf9fb45132e79365c9ef44","source":{"kind":"arxiv","id":"math/9301209","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9301209","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9301209v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9301209","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"UUVC4HX5QHV5","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"UUVC4HX5QHV5S3Q2","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"UUVC4HX5","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1993:UUVC4HX5QHV5S3Q2V76LL247PP","target":"record","payload":{"canonical_record":{"source":{"id":"math/9301209","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"ea37261527356fa26ecc2e637470b6b32456eb8c18312930c29bb4e9464b68e6","abstract_canon_sha256":"047fe888b383cc31616a2fb208a2cfc62e4c146fd5c38ba18e2b70878c12064b"},"schema_version":"1.0"},"canonical_sha256":"a52a2e1efd81ebd96e1aaffcb5eb9f7beca96fbf60cf9fb45132e79365c9ef44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:52.379158Z","signature_b64":"V7TQEa7jx3MN2/NGnrx+5a24+lomAkqyez5tX5frsxLQz0Xru78T4LTpbNcUTD6zQPAmVDjkavo7IQCuMDSoAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a52a2e1efd81ebd96e1aaffcb5eb9f7beca96fbf60cf9fb45132e79365c9ef44","last_reissued_at":"2026-05-18T01:05:52.378613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:52.378613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9301209","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AXGohSMr6UYbF70eZKpQmZWR/5yYoDmgn6k6bjIYVNswYrcFjxe8sbAj0poQA2cYJ/Zq1VMKjL/HCpaH6u4/Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T06:35:48.326690Z"},"content_sha256":"e2657f72556477389274b19a2acbce9a8bf5ed05abb6fca69775be6c513ac23e","schema_version":"1.0","event_id":"sha256:e2657f72556477389274b19a2acbce9a8bf5ed05abb6fca69775be6c513ac23e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1993:UUVC4HX5QHV5S3Q2V76LL247PP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Borel partitions of infinite subtrees of a perfect tree","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Alain Louveau, Boban Veli\\v{c}kovi\\'c, Saharon Shelah","submitted_at":"1993-01-15T00:00:00Z","abstract_excerpt":"A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n >= 3 is false. Let us identify the reals with 2^omega ordered by the lexicographical ordering and define for distinct x,y in 2^omega, D(x,y) to be the least n such that x(n) not= y(n). Let the type of an increasing n-tuple {x_0, ... x_{n-1}}_< be the ordering <^* on {0, ...,n-2} defined by i<^*j iff D(x_i,x_{i+1})< D(x_j,x_{j+1}). Galvin proved that for any Borel col"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9301209","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QLvthRBJQ9IeAlx+3TNhOo/xiBV00WtzpxTuOdHD8ukfVGhK98azlz5VqeA8khHcW+HmEMsiH3vtiaCIop8CDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T06:35:48.327026Z"},"content_sha256":"5f58ad781a39bb8c6273bc96e7a85702b4d3ce0772d34261b2e7744d29e3ab84","schema_version":"1.0","event_id":"sha256:5f58ad781a39bb8c6273bc96e7a85702b4d3ce0772d34261b2e7744d29e3ab84"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UUVC4HX5QHV5S3Q2V76LL247PP/bundle.json","state_url":"https://pith.science/pith/UUVC4HX5QHV5S3Q2V76LL247PP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UUVC4HX5QHV5S3Q2V76LL247PP/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":"2026-06-27T06:35:48Z","links":{"resolver":"https://pith.science/pith/UUVC4HX5QHV5S3Q2V76LL247PP","bundle":"https://pith.science/pith/UUVC4HX5QHV5S3Q2V76LL247PP/bundle.json","state":"https://pith.science/pith/UUVC4HX5QHV5S3Q2V76LL247PP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UUVC4HX5QHV5S3Q2V76LL247PP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1993:UUVC4HX5QHV5S3Q2V76LL247PP","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":"047fe888b383cc31616a2fb208a2cfc62e4c146fd5c38ba18e2b70878c12064b","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","title_canon_sha256":"ea37261527356fa26ecc2e637470b6b32456eb8c18312930c29bb4e9464b68e6"},"schema_version":"1.0","source":{"id":"math/9301209","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9301209","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9301209v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9301209","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"UUVC4HX5QHV5","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"UUVC4HX5QHV5S3Q2","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"UUVC4HX5","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:5f58ad781a39bb8c6273bc96e7a85702b4d3ce0772d34261b2e7744d29e3ab84","target":"graph","created_at":"2026-05-18T01:05:52Z","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 theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n >= 3 is false. Let us identify the reals with 2^omega ordered by the lexicographical ordering and define for distinct x,y in 2^omega, D(x,y) to be the least n such that x(n) not= y(n). Let the type of an increasing n-tuple {x_0, ... x_{n-1}}_< be the ordering <^* on {0, ...,n-2} defined by i<^*j iff D(x_i,x_{i+1})< D(x_j,x_{j+1}). Galvin proved that for any Borel col","authors_text":"Alain Louveau, Boban Veli\\v{c}kovi\\'c, Saharon Shelah","cross_cats":[],"headline":"","license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","title":"Borel partitions of infinite subtrees of a perfect tree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9301209","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:e2657f72556477389274b19a2acbce9a8bf5ed05abb6fca69775be6c513ac23e","target":"record","created_at":"2026-05-18T01:05:52Z","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":"047fe888b383cc31616a2fb208a2cfc62e4c146fd5c38ba18e2b70878c12064b","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1993-01-15T00:00:00Z","title_canon_sha256":"ea37261527356fa26ecc2e637470b6b32456eb8c18312930c29bb4e9464b68e6"},"schema_version":"1.0","source":{"id":"math/9301209","kind":"arxiv","version":1}},"canonical_sha256":"a52a2e1efd81ebd96e1aaffcb5eb9f7beca96fbf60cf9fb45132e79365c9ef44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a52a2e1efd81ebd96e1aaffcb5eb9f7beca96fbf60cf9fb45132e79365c9ef44","first_computed_at":"2026-05-18T01:05:52.378613Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:52.378613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V7TQEa7jx3MN2/NGnrx+5a24+lomAkqyez5tX5frsxLQz0Xru78T4LTpbNcUTD6zQPAmVDjkavo7IQCuMDSoAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:52.379158Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9301209","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e2657f72556477389274b19a2acbce9a8bf5ed05abb6fca69775be6c513ac23e","sha256:5f58ad781a39bb8c6273bc96e7a85702b4d3ce0772d34261b2e7744d29e3ab84"],"state_sha256":"65c24ee8f2aeab2365490363245b6b3d97c549cd95d865059237d4ffc51b1160"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ENuBvhxSINzb8RHvV/0bSzKe1L4A65O+KeoDHlQ0Y4+pBwNVuvcvj5cx3KAqrA78IP2tbTIlDqUOYgMvVt7cCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T06:35:48.328832Z","bundle_sha256":"48ce55c8725933c0ade0b4eac5ceea7ed741489ca051a633e41d91f9a13365ad"}}