{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:EEAW2DWFNDHWTHT7MEG2LBUSAT","short_pith_number":"pith:EEAW2DWF","canonical_record":{"source":{"id":"1805.06055","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T22:32:23Z","cross_cats_sorted":[],"title_canon_sha256":"03c3c972932bf10564a538a7abb7a4b7b490c25fa6261fb982ddc9096135be82","abstract_canon_sha256":"44d7d4c9f41aa5a3b6a1f7c244db938195944a707f7803b8cc816748d4ba7008"},"schema_version":"1.0"},"canonical_sha256":"21016d0ec568cf699e7f610da5869204e7cd1c689e3292bf3baf0897d5e6f41b","source":{"kind":"arxiv","id":"1805.06055","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.06055","created_at":"2026-05-18T00:15:49Z"},{"alias_kind":"arxiv_version","alias_value":"1805.06055v1","created_at":"2026-05-18T00:15:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.06055","created_at":"2026-05-18T00:15:49Z"},{"alias_kind":"pith_short_12","alias_value":"EEAW2DWFNDHW","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EEAW2DWFNDHWTHT7","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EEAW2DWF","created_at":"2026-05-18T12:32:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:EEAW2DWFNDHWTHT7MEG2LBUSAT","target":"record","payload":{"canonical_record":{"source":{"id":"1805.06055","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T22:32:23Z","cross_cats_sorted":[],"title_canon_sha256":"03c3c972932bf10564a538a7abb7a4b7b490c25fa6261fb982ddc9096135be82","abstract_canon_sha256":"44d7d4c9f41aa5a3b6a1f7c244db938195944a707f7803b8cc816748d4ba7008"},"schema_version":"1.0"},"canonical_sha256":"21016d0ec568cf699e7f610da5869204e7cd1c689e3292bf3baf0897d5e6f41b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:49.028440Z","signature_b64":"AdeHckfCKC+5/ukLTJcU3zN72qPCJ9v1nMH8s2a50vqXcUoWr2UFFdaVuiS4VfQRz8CuX4ydGewVXeqak5hcDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21016d0ec568cf699e7f610da5869204e7cd1c689e3292bf3baf0897d5e6f41b","last_reissued_at":"2026-05-18T00:15:49.027742Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:49.027742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.06055","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":"2026-05-18T00:15:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2YMXXtuI2BYIO5RMVFjBf81QkhNATb1PpZSgztsBdsVpXSljP85hInAAmQxTkR2S8w3W8pROYblZS5aP5ySEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:50:16.310078Z"},"content_sha256":"4242fa56c5a3deda78ec51ab85d276c0fa6a986571027a5a57c83dbc08b0d445","schema_version":"1.0","event_id":"sha256:4242fa56c5a3deda78ec51ab85d276c0fa6a986571027a5a57c83dbc08b0d445"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:EEAW2DWFNDHWTHT7MEG2LBUSAT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Hadwiger-Nelson problem with two forbidden distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Ismailescu, Geoffrey Exoo","submitted_at":"2018-05-15T22:32:23Z","abstract_excerpt":"In 1950 Edward Nelson asked the following simple-sounding question:\n  \\emph{How many colors are needed to color the Euclidean plane $\\mathbb{E}^2$ such that no two points distance $1$ apart are identically colored?}\n  We say that $1$ is a \\emph{forbidden} distance. For many years, we only knew that the answer was $4$, $5$, $6$, or $7$. In a recent breakthrough, de Grey \\cite{degrey} proved that at least five colors are necessary.\n  In this paper we consider a related problem in which we require \\emph{two} forbidden distances, $1$ and $d$. In other words, for a given positive number $d\\neq 1$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06055","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":"2026-05-18T00:15:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LZ0JJoqYay4eVUEvPXjW+vPmpeeiaKuwiUSV7HEEXk/4LORfaGEKVegWPV2z9rQCCMvff+KAYKTFDWTOTb9wBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:50:16.310431Z"},"content_sha256":"9195023249d7f70e0ed170cb69284f158f7f9077f1591b8bcf01b60959c1e49d","schema_version":"1.0","event_id":"sha256:9195023249d7f70e0ed170cb69284f158f7f9077f1591b8bcf01b60959c1e49d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EEAW2DWFNDHWTHT7MEG2LBUSAT/bundle.json","state_url":"https://pith.science/pith/EEAW2DWFNDHWTHT7MEG2LBUSAT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EEAW2DWFNDHWTHT7MEG2LBUSAT/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":"2026-06-25T12:50:16Z","links":{"resolver":"https://pith.science/pith/EEAW2DWFNDHWTHT7MEG2LBUSAT","bundle":"https://pith.science/pith/EEAW2DWFNDHWTHT7MEG2LBUSAT/bundle.json","state":"https://pith.science/pith/EEAW2DWFNDHWTHT7MEG2LBUSAT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EEAW2DWFNDHWTHT7MEG2LBUSAT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:EEAW2DWFNDHWTHT7MEG2LBUSAT","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":"44d7d4c9f41aa5a3b6a1f7c244db938195944a707f7803b8cc816748d4ba7008","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T22:32:23Z","title_canon_sha256":"03c3c972932bf10564a538a7abb7a4b7b490c25fa6261fb982ddc9096135be82"},"schema_version":"1.0","source":{"id":"1805.06055","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.06055","created_at":"2026-05-18T00:15:49Z"},{"alias_kind":"arxiv_version","alias_value":"1805.06055v1","created_at":"2026-05-18T00:15:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.06055","created_at":"2026-05-18T00:15:49Z"},{"alias_kind":"pith_short_12","alias_value":"EEAW2DWFNDHW","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EEAW2DWFNDHWTHT7","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EEAW2DWF","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:9195023249d7f70e0ed170cb69284f158f7f9077f1591b8bcf01b60959c1e49d","target":"graph","created_at":"2026-05-18T00:15:49Z","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":"In 1950 Edward Nelson asked the following simple-sounding question:\n  \\emph{How many colors are needed to color the Euclidean plane $\\mathbb{E}^2$ such that no two points distance $1$ apart are identically colored?}\n  We say that $1$ is a \\emph{forbidden} distance. For many years, we only knew that the answer was $4$, $5$, $6$, or $7$. In a recent breakthrough, de Grey \\cite{degrey} proved that at least five colors are necessary.\n  In this paper we consider a related problem in which we require \\emph{two} forbidden distances, $1$ and $d$. In other words, for a given positive number $d\\neq 1$, ","authors_text":"Dan Ismailescu, Geoffrey Exoo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T22:32:23Z","title":"The Hadwiger-Nelson problem with two forbidden distances"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06055","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":"sha256:4242fa56c5a3deda78ec51ab85d276c0fa6a986571027a5a57c83dbc08b0d445","target":"record","created_at":"2026-05-18T00:15:49Z","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":"44d7d4c9f41aa5a3b6a1f7c244db938195944a707f7803b8cc816748d4ba7008","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T22:32:23Z","title_canon_sha256":"03c3c972932bf10564a538a7abb7a4b7b490c25fa6261fb982ddc9096135be82"},"schema_version":"1.0","source":{"id":"1805.06055","kind":"arxiv","version":1}},"canonical_sha256":"21016d0ec568cf699e7f610da5869204e7cd1c689e3292bf3baf0897d5e6f41b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"21016d0ec568cf699e7f610da5869204e7cd1c689e3292bf3baf0897d5e6f41b","first_computed_at":"2026-05-18T00:15:49.027742Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:15:49.027742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AdeHckfCKC+5/ukLTJcU3zN72qPCJ9v1nMH8s2a50vqXcUoWr2UFFdaVuiS4VfQRz8CuX4ydGewVXeqak5hcDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:15:49.028440Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.06055","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4242fa56c5a3deda78ec51ab85d276c0fa6a986571027a5a57c83dbc08b0d445","sha256:9195023249d7f70e0ed170cb69284f158f7f9077f1591b8bcf01b60959c1e49d"],"state_sha256":"9e7d621bbe0a50fdceb2252e405a907a34ecac8bd11ebba7337175ffbe209541"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FEcJ7xvMwbshigRWzCbBDXWdQynw7O6L8qWrq//HVhwG/9vPD8C4jAAPcj+zBCYbdtMes0zldiJm3w/RqhlICw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T12:50:16.312285Z","bundle_sha256":"919cda059502f613b14cb1578fd957ff3a107da9719ff54ddf4ec7ceab2693bd"}}