{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:TR5HEYRMKDLQRYO57TLZLDC5DK","short_pith_number":"pith:TR5HEYRM","canonical_record":{"source":{"id":"0710.3877","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-10-20T21:47:51Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d7a0dd9019304cbd9d19b0e521c766471bf7f5ab799d9f804bdec07d83e6cd2e","abstract_canon_sha256":"5970cf09343569ff16d4ea8fd0a946e8f13ed4b73d4320c93c5ee1614e884256"},"schema_version":"1.0"},"canonical_sha256":"9c7a72622c50d708e1ddfcd7958c5d1ab3126452b822b168f3b6fc6754f47b0a","source":{"kind":"arxiv","id":"0710.3877","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.3877","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"arxiv_version","alias_value":"0710.3877v1","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.3877","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"pith_short_12","alias_value":"TR5HEYRMKDLQ","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"pith_short_16","alias_value":"TR5HEYRMKDLQRYO5","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"pith_short_8","alias_value":"TR5HEYRM","created_at":"2026-07-04T15:05:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:TR5HEYRMKDLQRYO57TLZLDC5DK","target":"record","payload":{"canonical_record":{"source":{"id":"0710.3877","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-10-20T21:47:51Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d7a0dd9019304cbd9d19b0e521c766471bf7f5ab799d9f804bdec07d83e6cd2e","abstract_canon_sha256":"5970cf09343569ff16d4ea8fd0a946e8f13ed4b73d4320c93c5ee1614e884256"},"schema_version":"1.0"},"canonical_sha256":"9c7a72622c50d708e1ddfcd7958c5d1ab3126452b822b168f3b6fc6754f47b0a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:05:04.726618Z","signature_b64":"1+CdlyN/eeqyFmSSRec4YNJZh6kJAvJb/ZAD06H9lWRN98HkJ+kdnRwfbl8SRmfXpcb9JguMvAxiAaIBGVVjCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c7a72622c50d708e1ddfcd7958c5d1ab3126452b822b168f3b6fc6754f47b0a","last_reissued_at":"2026-07-04T15:05:04.726219Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:05:04.726219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0710.3877","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":"2026-07-04T15:05:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1iMyFk+tQAMG7btA72NtTN39Y3RyopetNSfmzVmFqnV8xatZ51tRXd2ESQ9mBr1TqtslV+BK9PFuF0VKCQnHCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T15:15:28.293242Z"},"content_sha256":"b47e7a00acf2aabcf796125496cfb8798d5403c0ab70dee513d89007ab817661","schema_version":"1.0","event_id":"sha256:b47e7a00acf2aabcf796125496cfb8798d5403c0ab70dee513d89007ab817661"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:TR5HEYRMKDLQRYO57TLZLDC5DK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quasirandom groups","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"W. T. Gowers","submitted_at":"2007-10-20T21:47:51Z","abstract_excerpt":"Babai and S\\'os have asked whether there exists a constant c>0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy=z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.3877","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0710.3877/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T15:05:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JwzIfDAdz+6r9jVvtxNMYnAbskvC3lXjxSMwCSY4udA7/2M+685YaL35FxDBxQRLtpIh0hjGyKnWdtF6JkyyBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T15:15:28.294384Z"},"content_sha256":"9bd3469458c750f214ca6b2c4f0a1e8b25a4e72dde4108d6d9209a14e5f40b61","schema_version":"1.0","event_id":"sha256:9bd3469458c750f214ca6b2c4f0a1e8b25a4e72dde4108d6d9209a14e5f40b61"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TR5HEYRMKDLQRYO57TLZLDC5DK/bundle.json","state_url":"https://pith.science/pith/TR5HEYRMKDLQRYO57TLZLDC5DK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TR5HEYRMKDLQRYO57TLZLDC5DK/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":"2026-08-21T15:15:28Z","links":{"resolver":"https://pith.science/pith/TR5HEYRMKDLQRYO57TLZLDC5DK","bundle":"https://pith.science/pith/TR5HEYRMKDLQRYO57TLZLDC5DK/bundle.json","state":"https://pith.science/pith/TR5HEYRMKDLQRYO57TLZLDC5DK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TR5HEYRMKDLQRYO57TLZLDC5DK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:TR5HEYRMKDLQRYO57TLZLDC5DK","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":"5970cf09343569ff16d4ea8fd0a946e8f13ed4b73d4320c93c5ee1614e884256","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.CO","submitted_at":"2007-10-20T21:47:51Z","title_canon_sha256":"d7a0dd9019304cbd9d19b0e521c766471bf7f5ab799d9f804bdec07d83e6cd2e"},"schema_version":"1.0","source":{"id":"0710.3877","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.3877","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"arxiv_version","alias_value":"0710.3877v1","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.3877","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"pith_short_12","alias_value":"TR5HEYRMKDLQ","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"pith_short_16","alias_value":"TR5HEYRMKDLQRYO5","created_at":"2026-07-04T15:05:04Z"},{"alias_kind":"pith_short_8","alias_value":"TR5HEYRM","created_at":"2026-07-04T15:05:04Z"}],"graph_snapshots":[{"event_id":"sha256:9bd3469458c750f214ca6b2c4f0a1e8b25a4e72dde4108d6d9209a14e5f40b61","target":"graph","created_at":"2026-07-04T15:05:04Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0710.3877/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Babai and S\\'os have asked whether there exists a constant c>0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy=z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.","authors_text":"W. T. Gowers","cross_cats":["math.GR"],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2007-10-20T21:47:51Z","title":"Quasirandom groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.3877","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:b47e7a00acf2aabcf796125496cfb8798d5403c0ab70dee513d89007ab817661","target":"record","created_at":"2026-07-04T15:05:04Z","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":"5970cf09343569ff16d4ea8fd0a946e8f13ed4b73d4320c93c5ee1614e884256","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.CO","submitted_at":"2007-10-20T21:47:51Z","title_canon_sha256":"d7a0dd9019304cbd9d19b0e521c766471bf7f5ab799d9f804bdec07d83e6cd2e"},"schema_version":"1.0","source":{"id":"0710.3877","kind":"arxiv","version":1}},"canonical_sha256":"9c7a72622c50d708e1ddfcd7958c5d1ab3126452b822b168f3b6fc6754f47b0a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9c7a72622c50d708e1ddfcd7958c5d1ab3126452b822b168f3b6fc6754f47b0a","first_computed_at":"2026-07-04T15:05:04.726219Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:05:04.726219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1+CdlyN/eeqyFmSSRec4YNJZh6kJAvJb/ZAD06H9lWRN98HkJ+kdnRwfbl8SRmfXpcb9JguMvAxiAaIBGVVjCg==","signature_status":"signed_v1","signed_at":"2026-07-04T15:05:04.726618Z","signed_message":"canonical_sha256_bytes"},"source_id":"0710.3877","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b47e7a00acf2aabcf796125496cfb8798d5403c0ab70dee513d89007ab817661","sha256:9bd3469458c750f214ca6b2c4f0a1e8b25a4e72dde4108d6d9209a14e5f40b61"],"state_sha256":"4ed7d78c4f6173f9ef4d12dca2c6118199dea63f26c848e0fbc3745d534a9253"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ab2/RLWmZK/zrwuaDMMXmLr58FpoVBfxD6tN3F7I+e1+DhsdU5HUdGgZ15XhHErjh8GsQJtdt7VTOKwsNhkDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T15:15:28.299646Z","bundle_sha256":"9b7665bcdbbf9bf6f1c7217c82b5c3a70c7db49434ca62013387afed95b2cd50"}}