{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:RQ63PTRVAF2EUQ4CSCGXQX5ICJ","short_pith_number":"pith:RQ63PTRV","schema_version":"1.0","canonical_sha256":"8c3db7ce3501744a4382908d785fa812514c54a82cba76092f858d72f5de1641","source":{"kind":"arxiv","id":"1005.3750","version":2},"attestation_state":"computed","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{"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1005.3750","created_at":"2026-05-18T03:41:01.630794+00:00"},{"alias_kind":"arxiv_version","alias_value":"1005.3750v2","created_at":"2026-05-18T03:41:01.630794+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.3750","created_at":"2026-05-18T03:41:01.630794+00:00"},{"alias_kind":"pith_short_12","alias_value":"RQ63PTRVAF2E","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_16","alias_value":"RQ63PTRVAF2EUQ4C","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_8","alias_value":"RQ63PTRV","created_at":"2026-05-18T12:26:13.927090+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ","json":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ.json","graph_json":"https://pith.science/api/pith-number/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/graph.json","events_json":"https://pith.science/api/pith-number/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/events.json","paper":"https://pith.science/paper/RQ63PTRV"},"agent_actions":{"view_html":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ","download_json":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ.json","view_paper":"https://pith.science/paper/RQ63PTRV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1005.3750&json=true","fetch_graph":"https://pith.science/api/pith-number/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/graph.json","fetch_events":"https://pith.science/api/pith-number/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/action/storage_attestation","attest_author":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/action/author_attestation","sign_citation":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/action/citation_signature","submit_replication":"https://pith.science/pith/RQ63PTRVAF2EUQ4CSCGXQX5ICJ/action/replication_record"}},"created_at":"2026-05-18T03:41:01.630794+00:00","updated_at":"2026-05-18T03:41:01.630794+00:00"}