{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:CA6ST2DDQQSR73UTH3HHOZCLOP","short_pith_number":"pith:CA6ST2DD","canonical_record":{"source":{"id":"1211.5771","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-25T16:09:32Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2c855bddd644014a8322cb72f78d4d8763842d8da4293036c9b6b5f22db21590","abstract_canon_sha256":"f4ffd098d5528a5cc1f27107131211eda2451ac106871228406a04aa3ba7a5d8"},"schema_version":"1.0"},"canonical_sha256":"103d29e86384251fee933ece77644b73e17b99e338cbd37aa0494d9d4a9ad697","source":{"kind":"arxiv","id":"1211.5771","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5771","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5771v2","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5771","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"pith_short_12","alias_value":"CA6ST2DDQQSR","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CA6ST2DDQQSR73UT","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CA6ST2DD","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:CA6ST2DDQQSR73UTH3HHOZCLOP","target":"record","payload":{"canonical_record":{"source":{"id":"1211.5771","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-25T16:09:32Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2c855bddd644014a8322cb72f78d4d8763842d8da4293036c9b6b5f22db21590","abstract_canon_sha256":"f4ffd098d5528a5cc1f27107131211eda2451ac106871228406a04aa3ba7a5d8"},"schema_version":"1.0"},"canonical_sha256":"103d29e86384251fee933ece77644b73e17b99e338cbd37aa0494d9d4a9ad697","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:43.114401Z","signature_b64":"RnepOjxeRfjdmr0VjuoRNRnXvl5oM6bp5VRzYF1aE73sty1n+kGqS7z/dZUhOvQunJ2zHbYJv457TDbutpB0Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"103d29e86384251fee933ece77644b73e17b99e338cbd37aa0494d9d4a9ad697","last_reissued_at":"2026-05-18T03:39:43.113899Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:43.113899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.5771","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":"2026-05-18T03:39:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nb6l7q9GWJMc1dPntVsxI/2gG/9cjs7wRSu5FUZJ5a99VI5zq4HgcmZFVnL/TL4nqSHjoRhMB6wIhbPbWPEVCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T20:02:53.847097Z"},"content_sha256":"6d4c33f87fd51242fb780b4f80d24b17801ba3bcd9542c4478252b521ee47130","schema_version":"1.0","event_id":"sha256:6d4c33f87fd51242fb780b4f80d24b17801ba3bcd9542c4478252b521ee47130"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:CA6ST2DDQQSR73UTH3HHOZCLOP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Capturing Forms in Dense Subsets of Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Brandon Hanson","submitted_at":"2012-11-25T16:09:32Z","abstract_excerpt":"An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\\mathbb{N}\\to\\{1,...,r\\}$ of the natural numbers, there exist $x,y\\in \\mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More generally, one could replace $x+y$ with a binary linear form and $xy$ with a binary quadratic form. In this paper we examine the analogous problem in a finite field $\\mathbb{F}_q$. Specifically, given a linear form $L$ and a quadratic from $Q$ in two variables, we provide estimates on the necessary size of $A\\subset \\mathbb{F}_q$ to guarantee that $L(x,y)$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5771","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":"2026-05-18T03:39:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NR7o8OVcxILLeMZv1qV5GnK2XmbY4G+Rsm8J5ANB28r9xwtxpO9rn+xz6aycBgLakNoWmNALZ/0d+sjRMx9pDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T20:02:53.847460Z"},"content_sha256":"aa24d11d439de6a55a5fcad239ddf8bc1ca8636e80b9ba613b70b3063ca47757","schema_version":"1.0","event_id":"sha256:aa24d11d439de6a55a5fcad239ddf8bc1ca8636e80b9ba613b70b3063ca47757"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CA6ST2DDQQSR73UTH3HHOZCLOP/bundle.json","state_url":"https://pith.science/pith/CA6ST2DDQQSR73UTH3HHOZCLOP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CA6ST2DDQQSR73UTH3HHOZCLOP/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":"2026-06-26T20:02:53Z","links":{"resolver":"https://pith.science/pith/CA6ST2DDQQSR73UTH3HHOZCLOP","bundle":"https://pith.science/pith/CA6ST2DDQQSR73UTH3HHOZCLOP/bundle.json","state":"https://pith.science/pith/CA6ST2DDQQSR73UTH3HHOZCLOP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CA6ST2DDQQSR73UTH3HHOZCLOP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CA6ST2DDQQSR73UTH3HHOZCLOP","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":"f4ffd098d5528a5cc1f27107131211eda2451ac106871228406a04aa3ba7a5d8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-25T16:09:32Z","title_canon_sha256":"2c855bddd644014a8322cb72f78d4d8763842d8da4293036c9b6b5f22db21590"},"schema_version":"1.0","source":{"id":"1211.5771","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5771","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5771v2","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5771","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"pith_short_12","alias_value":"CA6ST2DDQQSR","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CA6ST2DDQQSR73UT","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CA6ST2DD","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:aa24d11d439de6a55a5fcad239ddf8bc1ca8636e80b9ba613b70b3063ca47757","target":"graph","created_at":"2026-05-18T03:39:43Z","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":"An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\\mathbb{N}\\to\\{1,...,r\\}$ of the natural numbers, there exist $x,y\\in \\mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More generally, one could replace $x+y$ with a binary linear form and $xy$ with a binary quadratic form. In this paper we examine the analogous problem in a finite field $\\mathbb{F}_q$. Specifically, given a linear form $L$ and a quadratic from $Q$ in two variables, we provide estimates on the necessary size of $A\\subset \\mathbb{F}_q$ to guarantee that $L(x,y)$ and ","authors_text":"Brandon Hanson","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-25T16:09:32Z","title":"Capturing Forms in Dense Subsets of Finite Fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5771","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":"sha256:6d4c33f87fd51242fb780b4f80d24b17801ba3bcd9542c4478252b521ee47130","target":"record","created_at":"2026-05-18T03:39:43Z","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":"f4ffd098d5528a5cc1f27107131211eda2451ac106871228406a04aa3ba7a5d8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-25T16:09:32Z","title_canon_sha256":"2c855bddd644014a8322cb72f78d4d8763842d8da4293036c9b6b5f22db21590"},"schema_version":"1.0","source":{"id":"1211.5771","kind":"arxiv","version":2}},"canonical_sha256":"103d29e86384251fee933ece77644b73e17b99e338cbd37aa0494d9d4a9ad697","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"103d29e86384251fee933ece77644b73e17b99e338cbd37aa0494d9d4a9ad697","first_computed_at":"2026-05-18T03:39:43.113899Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:43.113899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RnepOjxeRfjdmr0VjuoRNRnXvl5oM6bp5VRzYF1aE73sty1n+kGqS7z/dZUhOvQunJ2zHbYJv457TDbutpB0Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:43.114401Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.5771","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d4c33f87fd51242fb780b4f80d24b17801ba3bcd9542c4478252b521ee47130","sha256:aa24d11d439de6a55a5fcad239ddf8bc1ca8636e80b9ba613b70b3063ca47757"],"state_sha256":"e95a775e36deba4b1f30695104701b58626c4dbbdf79c269db30406992f36111"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FCzak/qlrf05kiwThJJ/+NnPj4bdr5o+CF7HpTGKI+2rw/bxXyd7BnDlvjD4WMnry3iCKPM98K3U97+Yog/tCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T20:02:53.849354Z","bundle_sha256":"cea6d0098ee609a00325090257e5acebf7ba2bbf663829457b5f176e3a3084c7"}}