{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:RQ63PTRVAF2EUQ4CSCGXQX5ICJ","short_pith_number":"pith:RQ63PTRV","canonical_record":{"source":{"id":"1005.3750","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-20T16:12:40Z","cross_cats_sorted":[],"title_canon_sha256":"a106cc2fbd9c5846f8eb548c043a42a18865f31d7ee48694b6f403db8badea5f","abstract_canon_sha256":"e8c1c681bb8a87df63632f520f8fd3c29251e5f7b1744b94c123d42cb0caf436"},"schema_version":"1.0"},"canonical_sha256":"8c3db7ce3501744a4382908d785fa812514c54a82cba76092f858d72f5de1641","source":{"kind":"arxiv","id":"1005.3750","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.3750","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1005.3750v2","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.3750","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"RQ63PTRVAF2E","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RQ63PTRVAF2EUQ4C","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RQ63PTRV","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:RQ63PTRVAF2EUQ4CSCGXQX5ICJ","target":"record","payload":{"canonical_record":{"source":{"id":"1005.3750","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-20T16:12:40Z","cross_cats_sorted":[],"title_canon_sha256":"a106cc2fbd9c5846f8eb548c043a42a18865f31d7ee48694b6f403db8badea5f","abstract_canon_sha256":"e8c1c681bb8a87df63632f520f8fd3c29251e5f7b1744b94c123d42cb0caf436"},"schema_version":"1.0"},"canonical_sha256":"8c3db7ce3501744a4382908d785fa812514c54a82cba76092f858d72f5de1641","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:01.631373Z","signature_b64":"V/tkwtB/cXgwGVsCNmjz3RRtPGHXP9vbF21J5nldxdV+5YlBfc1v/dPE9/hB2XVCletziJM7InLJwFBZmVjJBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c3db7ce3501744a4382908d785fa812514c54a82cba76092f858d72f5de1641","last_reissued_at":"2026-05-18T03:41:01.630673Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:01.630673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.3750","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":"2026-05-18T03:41:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XddWufUnQ2Vs0dm6d6NrDuwXrBBSfcK4Oi5mITLqjo8OSGIsNqrmxauRrlphSOQf0hRgOOSvntNHX1ohOOh2DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:24:42.429250Z"},"content_sha256":"1d731281b30866e5b4b2540c6abdc43337a95fe40a1263e9d1cae0469498d12f","schema_version":"1.0","event_id":"sha256:1d731281b30866e5b4b2540c6abdc43337a95fe40a1263e9d1cae0469498d12f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:RQ63PTRVAF2EUQ4CSCGXQX5ICJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rectangle Free Coloring of Grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Charles Glover, Semmy Purewal, Stephen Fenner, William Gasarch","submitted_at":"2010-05-20T16:12:40Z","abstract_excerpt":"A two-dimensional \\emph{grid} is a set $\\Gnm = [n]\\times[m]$. A grid $\\Gnm$ is \\emph{$c$-colorable} if there is a function $\\chi_{n,m}: \\Gnm \\to [c]$ such that there are no rectangles with all four corners the same color. We address the following question: for which values of $n$ and $m$ is $\\Gnm$ $c$-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der Waerden's Theorem). We determine (1) \\emph{exactly} which grids are 2-colorable, (2) \\emph{exactly} which grids are 3-colorable, and (3) \\emph{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.3750","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":"2026-05-18T03:41:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yz7wFNCCrqJ2+lJsrZwdF7664jzGxjusNjfgKDE7x+SKqUHlUS6hGx+qvePWcA6U0MsvL2YM4Fsyg8I42f8CBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:24:42.429603Z"},"content_sha256":"10cc89d9f1ad88035b4ada693092a956654c8d5ba0ba87d2e55181d20fae9372","schema_version":"1.0","event_id":"sha256:10cc89d9f1ad88035b4ada693092a956654c8d5ba0ba87d2e55181d20fae9372"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/bundle.json","state_url":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/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":"2026-05-28T10:24:42Z","links":{"resolver":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ","bundle":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/bundle.json","state":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:RQ63PTRVAF2EUQ4CSCGXQX5ICJ","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":"e8c1c681bb8a87df63632f520f8fd3c29251e5f7b1744b94c123d42cb0caf436","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-20T16:12:40Z","title_canon_sha256":"a106cc2fbd9c5846f8eb548c043a42a18865f31d7ee48694b6f403db8badea5f"},"schema_version":"1.0","source":{"id":"1005.3750","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.3750","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1005.3750v2","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.3750","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"RQ63PTRVAF2E","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RQ63PTRVAF2EUQ4C","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RQ63PTRV","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:10cc89d9f1ad88035b4ada693092a956654c8d5ba0ba87d2e55181d20fae9372","target":"graph","created_at":"2026-05-18T03:41:01Z","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 two-dimensional \\emph{grid} is a set $\\Gnm = [n]\\times[m]$. A grid $\\Gnm$ is \\emph{$c$-colorable} if there is a function $\\chi_{n,m}: \\Gnm \\to [c]$ such that there are no rectangles with all four corners the same color. We address the following question: for which values of $n$ and $m$ is $\\Gnm$ $c$-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der Waerden's Theorem). We determine (1) \\emph{exactly} which grids are 2-colorable, (2) \\emph{exactly} which grids are 3-colorable, and (3) \\emph{","authors_text":"Charles Glover, Semmy Purewal, Stephen Fenner, William Gasarch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-20T16:12:40Z","title":"Rectangle Free Coloring of Grids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.3750","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":"sha256:1d731281b30866e5b4b2540c6abdc43337a95fe40a1263e9d1cae0469498d12f","target":"record","created_at":"2026-05-18T03:41:01Z","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":"e8c1c681bb8a87df63632f520f8fd3c29251e5f7b1744b94c123d42cb0caf436","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-20T16:12:40Z","title_canon_sha256":"a106cc2fbd9c5846f8eb548c043a42a18865f31d7ee48694b6f403db8badea5f"},"schema_version":"1.0","source":{"id":"1005.3750","kind":"arxiv","version":2}},"canonical_sha256":"8c3db7ce3501744a4382908d785fa812514c54a82cba76092f858d72f5de1641","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c3db7ce3501744a4382908d785fa812514c54a82cba76092f858d72f5de1641","first_computed_at":"2026-05-18T03:41:01.630673Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:01.630673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V/tkwtB/cXgwGVsCNmjz3RRtPGHXP9vbF21J5nldxdV+5YlBfc1v/dPE9/hB2XVCletziJM7InLJwFBZmVjJBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:01.631373Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.3750","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d731281b30866e5b4b2540c6abdc43337a95fe40a1263e9d1cae0469498d12f","sha256:10cc89d9f1ad88035b4ada693092a956654c8d5ba0ba87d2e55181d20fae9372"],"state_sha256":"06539959e5086a2e466a71f256f7ef4630b76852221853e8bf3280f2d1aaaa3f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9CaEtzX6jLMw8SyLhVMfpUqfIFaOBOavR/UR3/HICygG3I21i5f1D5rwiOH8wOkxxhCwNInMuVsKR/5rydJPDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T10:24:42.431540Z","bundle_sha256":"ea9765449a69253ca5c4b0a5047599035074a3764a20010ff4828e0f37bb36f6"}}