{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:ASXUBSNJUWH4HUGWDQCNE57QU6","short_pith_number":"pith:ASXUBSNJ","canonical_record":{"source":{"id":"2101.05914","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-01-15T00:02:57Z","cross_cats_sorted":[],"title_canon_sha256":"f5e4ea4a4e9dcca047880a0cb4f65793d715b3f81670886ebb83cbcbe79a830e","abstract_canon_sha256":"b28f217aae4bda9a4e6c9071b4cd4ddf1ce12b32d9c082da5b1640a67cc01854"},"schema_version":"1.0"},"canonical_sha256":"04af40c9a9a58fc3d0d61c04d277f0a7b6a94a6689f43a6046de32b4ac7bbdbe","source":{"kind":"arxiv","id":"2101.05914","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.05914","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"arxiv_version","alias_value":"2101.05914v2","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.05914","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"pith_short_12","alias_value":"ASXUBSNJUWH4","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"pith_short_16","alias_value":"ASXUBSNJUWH4HUGW","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"pith_short_8","alias_value":"ASXUBSNJ","created_at":"2026-07-05T03:07:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:ASXUBSNJUWH4HUGWDQCNE57QU6","target":"record","payload":{"canonical_record":{"source":{"id":"2101.05914","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-01-15T00:02:57Z","cross_cats_sorted":[],"title_canon_sha256":"f5e4ea4a4e9dcca047880a0cb4f65793d715b3f81670886ebb83cbcbe79a830e","abstract_canon_sha256":"b28f217aae4bda9a4e6c9071b4cd4ddf1ce12b32d9c082da5b1640a67cc01854"},"schema_version":"1.0"},"canonical_sha256":"04af40c9a9a58fc3d0d61c04d277f0a7b6a94a6689f43a6046de32b4ac7bbdbe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:07:04.519596Z","signature_b64":"GHe5iivL+4lYl1c6LLpSbubxdp5KfqfENq+kBS3g6iD6p0t6qWMjvlEE8iNgqa3oM4qvQ8t53ODM9jXDYvKyDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04af40c9a9a58fc3d0d61c04d277f0a7b6a94a6689f43a6046de32b4ac7bbdbe","last_reissued_at":"2026-07-05T03:07:04.519201Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:07:04.519201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2101.05914","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":"2026-07-05T03:07:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aBz062pxetn289j4PxsFLwCqyrCvxfoQuGJFyCvgpNCZkrtYmhFMSWmg2V86IYYinSALwExBf2tgR5utGnl9Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T04:52:47.780214Z"},"content_sha256":"ec328fede573d9ff4b8ce79e43195c77c68e90be61e7fbefe97ae365754c2581","schema_version":"1.0","event_id":"sha256:ec328fede573d9ff4b8ce79e43195c77c68e90be61e7fbefe97ae365754c2581"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:ASXUBSNJUWH4HUGWDQCNE57QU6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the number of sum-free triplets of sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Igor Araujo, J\\'ozsef Balogh, Ramon I. Garcia","submitted_at":"2021-01-15T00:02:57Z","abstract_excerpt":"We count the ordered sum-free triplets of subsets in the group $\\mathbb{Z}/p\\mathbb{Z}$, i.e., the triplets $(A,B,C)$ of sets $A,B,C \\subset \\mathbb{Z}/p\\mathbb{Z}$ for which the equation $a+b=c$ has no solution with $a\\in A$, $b \\in B$ and $c \\in C$. Our main theorem improves on a recent result by Semchankau, Shabanov, and Shkredov using a different and simpler method. Our proof relates previous results on the number of independent sets of regular graphs by Kahn, Perarnau and Perkins, and Csikv\\'ari to produce explicit estimates on smaller order terms. We also obtain estimates for the number "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.05914","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.05914/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":"2026-07-05T03:07:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4/uF6ea+75a5KvaKWJ5NseNKR+ivk0h6mkKjoEZHhe8MI3rLzba8/W2YyfALQ+4B11KrGlQaXkKe+PQitkFCBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T04:52:47.780845Z"},"content_sha256":"c447be386cbe81ed1e692bfcb74c3475e8c16b4cb86b9c742968ee680c119545","schema_version":"1.0","event_id":"sha256:c447be386cbe81ed1e692bfcb74c3475e8c16b4cb86b9c742968ee680c119545"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ASXUBSNJUWH4HUGWDQCNE57QU6/bundle.json","state_url":"https://pith.science/pith/ASXUBSNJUWH4HUGWDQCNE57QU6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ASXUBSNJUWH4HUGWDQCNE57QU6/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":"2026-07-07T04:52:47Z","links":{"resolver":"https://pith.science/pith/ASXUBSNJUWH4HUGWDQCNE57QU6","bundle":"https://pith.science/pith/ASXUBSNJUWH4HUGWDQCNE57QU6/bundle.json","state":"https://pith.science/pith/ASXUBSNJUWH4HUGWDQCNE57QU6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ASXUBSNJUWH4HUGWDQCNE57QU6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:ASXUBSNJUWH4HUGWDQCNE57QU6","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":"b28f217aae4bda9a4e6c9071b4cd4ddf1ce12b32d9c082da5b1640a67cc01854","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-01-15T00:02:57Z","title_canon_sha256":"f5e4ea4a4e9dcca047880a0cb4f65793d715b3f81670886ebb83cbcbe79a830e"},"schema_version":"1.0","source":{"id":"2101.05914","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.05914","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"arxiv_version","alias_value":"2101.05914v2","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.05914","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"pith_short_12","alias_value":"ASXUBSNJUWH4","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"pith_short_16","alias_value":"ASXUBSNJUWH4HUGW","created_at":"2026-07-05T03:07:04Z"},{"alias_kind":"pith_short_8","alias_value":"ASXUBSNJ","created_at":"2026-07-05T03:07:04Z"}],"graph_snapshots":[{"event_id":"sha256:c447be386cbe81ed1e692bfcb74c3475e8c16b4cb86b9c742968ee680c119545","target":"graph","created_at":"2026-07-05T03:07:04Z","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/2101.05914/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We count the ordered sum-free triplets of subsets in the group $\\mathbb{Z}/p\\mathbb{Z}$, i.e., the triplets $(A,B,C)$ of sets $A,B,C \\subset \\mathbb{Z}/p\\mathbb{Z}$ for which the equation $a+b=c$ has no solution with $a\\in A$, $b \\in B$ and $c \\in C$. Our main theorem improves on a recent result by Semchankau, Shabanov, and Shkredov using a different and simpler method. Our proof relates previous results on the number of independent sets of regular graphs by Kahn, Perarnau and Perkins, and Csikv\\'ari to produce explicit estimates on smaller order terms. We also obtain estimates for the number ","authors_text":"Igor Araujo, J\\'ozsef Balogh, Ramon I. Garcia","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-01-15T00:02:57Z","title":"On the number of sum-free triplets of sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.05914","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:ec328fede573d9ff4b8ce79e43195c77c68e90be61e7fbefe97ae365754c2581","target":"record","created_at":"2026-07-05T03:07:04Z","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":"b28f217aae4bda9a4e6c9071b4cd4ddf1ce12b32d9c082da5b1640a67cc01854","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-01-15T00:02:57Z","title_canon_sha256":"f5e4ea4a4e9dcca047880a0cb4f65793d715b3f81670886ebb83cbcbe79a830e"},"schema_version":"1.0","source":{"id":"2101.05914","kind":"arxiv","version":2}},"canonical_sha256":"04af40c9a9a58fc3d0d61c04d277f0a7b6a94a6689f43a6046de32b4ac7bbdbe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"04af40c9a9a58fc3d0d61c04d277f0a7b6a94a6689f43a6046de32b4ac7bbdbe","first_computed_at":"2026-07-05T03:07:04.519201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:07:04.519201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GHe5iivL+4lYl1c6LLpSbubxdp5KfqfENq+kBS3g6iD6p0t6qWMjvlEE8iNgqa3oM4qvQ8t53ODM9jXDYvKyDg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:07:04.519596Z","signed_message":"canonical_sha256_bytes"},"source_id":"2101.05914","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec328fede573d9ff4b8ce79e43195c77c68e90be61e7fbefe97ae365754c2581","sha256:c447be386cbe81ed1e692bfcb74c3475e8c16b4cb86b9c742968ee680c119545"],"state_sha256":"b851317489f4f2eeac6d74dc6f106a3fb78409f4f4dae0f738d397e6219bde3a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hTYFpvjj00Xo+yZiYtr3Dl0UswRJ1xCDc7TTJM0eMBeoA0OY/YyDAdAsnuHdjCHblmT0ey7N/VkfT8kS0QqPBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T04:52:47.784807Z","bundle_sha256":"0643634caa95f1f4cf156709dccff155ab19fbc8f3a8a43a1bfe8b495786e195"}}