{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:3AIZOJAM55DWPE2KT63BNCNVSY","short_pith_number":"pith:3AIZOJAM","canonical_record":{"source":{"id":"1812.09259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-20T13:10:21Z","cross_cats_sorted":[],"title_canon_sha256":"4d5b26744d6e156cedc1a61d90bd09b00eb0f31136ffef10f66a07d808714444","abstract_canon_sha256":"e14bc3adc05bc67dbd0796c1286f70990ee44ff81f991a0572f135d9a5f9a0c2"},"schema_version":"1.0"},"canonical_sha256":"d81197240cef4767934a9fb61689b59622297399fefd6e39c7607d6d539aa432","source":{"kind":"arxiv","id":"1812.09259","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.09259","created_at":"2026-05-17T23:57:44Z"},{"alias_kind":"arxiv_version","alias_value":"1812.09259v1","created_at":"2026-05-17T23:57:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.09259","created_at":"2026-05-17T23:57:44Z"},{"alias_kind":"pith_short_12","alias_value":"3AIZOJAM55DW","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"3AIZOJAM55DWPE2K","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"3AIZOJAM","created_at":"2026-05-18T12:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:3AIZOJAM55DWPE2KT63BNCNVSY","target":"record","payload":{"canonical_record":{"source":{"id":"1812.09259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-20T13:10:21Z","cross_cats_sorted":[],"title_canon_sha256":"4d5b26744d6e156cedc1a61d90bd09b00eb0f31136ffef10f66a07d808714444","abstract_canon_sha256":"e14bc3adc05bc67dbd0796c1286f70990ee44ff81f991a0572f135d9a5f9a0c2"},"schema_version":"1.0"},"canonical_sha256":"d81197240cef4767934a9fb61689b59622297399fefd6e39c7607d6d539aa432","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:44.958712Z","signature_b64":"3+SGiJN7XCmROedRTUBObtoDnE/XlyswmgdhqGpiY3W+q1mMBAZSN+6ZN/EjrnVB/DMyRFdpwpP7ivIYiSR4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d81197240cef4767934a9fb61689b59622297399fefd6e39c7607d6d539aa432","last_reissued_at":"2026-05-17T23:57:44.958071Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:44.958071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.09259","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":"2026-05-17T23:57:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KYypbcQXnPgjWrWbrNNLJQXW/66qNwDmG+21Vayr6ihV0c/ak8iez3+l+20Js0PsWq9pHnbuoXS3c1tXMKgeCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:37:42.692714Z"},"content_sha256":"fd77320c8976a75790039d7771c945b4d3d53bf2f2bf01e5d002c0caf8de3d85","schema_version":"1.0","event_id":"sha256:fd77320c8976a75790039d7771c945b4d3d53bf2f2bf01e5d002c0caf8de3d85"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:3AIZOJAM55DWPE2KT63BNCNVSY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The complexity of solution-free sets of integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Keith J. Edwards, Steven D. Noble","submitted_at":"2018-12-20T13:10:21Z","abstract_excerpt":"Given a linear equation L, a set A of integers is L-free if A does not contain any non-trivial solutions to L. Meeks and Treglown showed that for certain kinds of linear equations, it is NP-complete to decide if a given set of integers contains a solution-free subset of a given size. Also, for equations involving three variables, they showed that the problem of determining the size of the largest solution-free subset is APX-hard, and that for two such equations (representing sum-free and progression-free sets), the problem of deciding if there is a solution-free subset with at least a specifie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.09259","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":"2026-05-17T23:57:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PGRtGEFk92F8QV8wdCQ9Fe+9SesavHI6Trc5nmuCcXyR+eRdPXjsLo9DTsk0p+xaULhEuqAK71RY4xiyvlvLAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:37:42.693188Z"},"content_sha256":"87c9afe89452e7ac2139d02e8d6962990ed7d6c6ab6247ddb1c38d82ccf7804b","schema_version":"1.0","event_id":"sha256:87c9afe89452e7ac2139d02e8d6962990ed7d6c6ab6247ddb1c38d82ccf7804b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3AIZOJAM55DWPE2KT63BNCNVSY/bundle.json","state_url":"https://pith.science/pith/3AIZOJAM55DWPE2KT63BNCNVSY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3AIZOJAM55DWPE2KT63BNCNVSY/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":"2026-06-24T06:37:42Z","links":{"resolver":"https://pith.science/pith/3AIZOJAM55DWPE2KT63BNCNVSY","bundle":"https://pith.science/pith/3AIZOJAM55DWPE2KT63BNCNVSY/bundle.json","state":"https://pith.science/pith/3AIZOJAM55DWPE2KT63BNCNVSY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3AIZOJAM55DWPE2KT63BNCNVSY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:3AIZOJAM55DWPE2KT63BNCNVSY","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":"e14bc3adc05bc67dbd0796c1286f70990ee44ff81f991a0572f135d9a5f9a0c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-20T13:10:21Z","title_canon_sha256":"4d5b26744d6e156cedc1a61d90bd09b00eb0f31136ffef10f66a07d808714444"},"schema_version":"1.0","source":{"id":"1812.09259","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.09259","created_at":"2026-05-17T23:57:44Z"},{"alias_kind":"arxiv_version","alias_value":"1812.09259v1","created_at":"2026-05-17T23:57:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.09259","created_at":"2026-05-17T23:57:44Z"},{"alias_kind":"pith_short_12","alias_value":"3AIZOJAM55DW","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"3AIZOJAM55DWPE2K","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"3AIZOJAM","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:87c9afe89452e7ac2139d02e8d6962990ed7d6c6ab6247ddb1c38d82ccf7804b","target":"graph","created_at":"2026-05-17T23:57:44Z","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":"Given a linear equation L, a set A of integers is L-free if A does not contain any non-trivial solutions to L. Meeks and Treglown showed that for certain kinds of linear equations, it is NP-complete to decide if a given set of integers contains a solution-free subset of a given size. Also, for equations involving three variables, they showed that the problem of determining the size of the largest solution-free subset is APX-hard, and that for two such equations (representing sum-free and progression-free sets), the problem of deciding if there is a solution-free subset with at least a specifie","authors_text":"Keith J. Edwards, Steven D. Noble","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-20T13:10:21Z","title":"The complexity of solution-free sets of integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.09259","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:fd77320c8976a75790039d7771c945b4d3d53bf2f2bf01e5d002c0caf8de3d85","target":"record","created_at":"2026-05-17T23:57:44Z","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":"e14bc3adc05bc67dbd0796c1286f70990ee44ff81f991a0572f135d9a5f9a0c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-20T13:10:21Z","title_canon_sha256":"4d5b26744d6e156cedc1a61d90bd09b00eb0f31136ffef10f66a07d808714444"},"schema_version":"1.0","source":{"id":"1812.09259","kind":"arxiv","version":1}},"canonical_sha256":"d81197240cef4767934a9fb61689b59622297399fefd6e39c7607d6d539aa432","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d81197240cef4767934a9fb61689b59622297399fefd6e39c7607d6d539aa432","first_computed_at":"2026-05-17T23:57:44.958071Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:44.958071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3+SGiJN7XCmROedRTUBObtoDnE/XlyswmgdhqGpiY3W+q1mMBAZSN+6ZN/EjrnVB/DMyRFdpwpP7ivIYiSR4Bg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:44.958712Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.09259","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fd77320c8976a75790039d7771c945b4d3d53bf2f2bf01e5d002c0caf8de3d85","sha256:87c9afe89452e7ac2139d02e8d6962990ed7d6c6ab6247ddb1c38d82ccf7804b"],"state_sha256":"6c491618bf653fdbe4e8647c0d056b533ea5f1d863ce629d6745627cb4d151cc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kCZGEo1jpX9RfDjfr8FR9eEM3S9XZ9Ol8uqNHVivLGUybRb4/ZpEwJpzfBGBWqijc3RhnXP4TVMqcCyv8allDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T06:37:42.696333Z","bundle_sha256":"efbdf7bb0d5dcd45248b17cf1dbac3e12017495f66eac0293251e490f688c6e7"}}