{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:4WGSR7WDJYDYNB5N4DNMCUWIYK","short_pith_number":"pith:4WGSR7WD","canonical_record":{"source":{"id":"2011.09402","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T17:00:01Z","cross_cats_sorted":[],"title_canon_sha256":"13d3cb37a204141cd0e60eece73b5a71e887904e9983dc1e0fb1026c87ca6da3","abstract_canon_sha256":"aef1c79d7d60ebeee027e9b421f51915966a9a756deae324c8ec3d75c01385e5"},"schema_version":"1.0"},"canonical_sha256":"e58d28fec34e078687ade0dac152c8c29b16e1e1ed051d668e7926d5472269b8","source":{"kind":"arxiv","id":"2011.09402","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.09402","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"arxiv_version","alias_value":"2011.09402v1","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.09402","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_12","alias_value":"4WGSR7WDJYDY","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_16","alias_value":"4WGSR7WDJYDYNB5N","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_8","alias_value":"4WGSR7WD","created_at":"2026-07-05T01:52:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:4WGSR7WDJYDYNB5N4DNMCUWIYK","target":"record","payload":{"canonical_record":{"source":{"id":"2011.09402","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T17:00:01Z","cross_cats_sorted":[],"title_canon_sha256":"13d3cb37a204141cd0e60eece73b5a71e887904e9983dc1e0fb1026c87ca6da3","abstract_canon_sha256":"aef1c79d7d60ebeee027e9b421f51915966a9a756deae324c8ec3d75c01385e5"},"schema_version":"1.0"},"canonical_sha256":"e58d28fec34e078687ade0dac152c8c29b16e1e1ed051d668e7926d5472269b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:52:43.655135Z","signature_b64":"LMpWojSqQAcKdbAu3R2gOc2RjKznDPBD+NHYBS6oMdQ1j8PdDtHhUT7fRgAaUNCiYKHHCB5FNIS8OSZwMuVJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e58d28fec34e078687ade0dac152c8c29b16e1e1ed051d668e7926d5472269b8","last_reissued_at":"2026-07-05T01:52:43.654732Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:52:43.654732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2011.09402","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":"2026-07-05T01:52:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"npeH4rj8F18VQILA7NOoTpfldz4zYAry1aaOGUEzGW7IpCWAhjiaUsMFTgN1IjfvLS1Fm3LGY92tGXY+DcoIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T09:42:22.951915Z"},"content_sha256":"c31d72f52987cb42b56c7b5084b8cae79b9cc0d55cb88dbccad1b7048b985275","schema_version":"1.0","event_id":"sha256:c31d72f52987cb42b56c7b5084b8cae79b9cc0d55cb88dbccad1b7048b985275"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:4WGSR7WDJYDYNB5N4DNMCUWIYK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on $k$-wise oddtown problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacques Verstra\\\"ete, Jason O'Neill","submitted_at":"2020-11-18T17:00:01Z","abstract_excerpt":"For integers $2 \\leq t \\leq k$, we consider a collection of $k$ set families $\\mathcal{A}_j: 1 \\leq j \\leq k$ where $\\mathcal{A}_j = \\{ A_{j,i} \\subseteq [n] : 1 \\leq i \\leq m \\}$ and $|A_{1, i_1} \\cap \\cdots \\cap A_{k,i_k}|$ is even if and only if at least $t$ of the $i_j$ are distinct. In this paper, we prove that $m =O(n^{ 1/ \\lfloor k/2 \\rfloor})$ when $t=k$ and $m = O( n^{1/(t-1)})$ when $2t-2 \\leq k$ and prove that both of these bounds are best possible. Specializing to the case where $\\mathcal{A} = \\mathcal{A}_1 = \\cdots = \\mathcal{A}_k$, we recover a variation of the classical oddtown "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.09402","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/2011.09402/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":"2026-07-05T01:52:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ZM0NXcP6lmOj4+jb3tub6u30Nov9PzfNoUBx2ZoOQo6IVQL3uSrpiGl1I6BMD6uck1LLEFJAPzThJHz9obxDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T09:42:22.952295Z"},"content_sha256":"cba1cbf450c75174da8d14a84e753cfa700d7c7eb71d743973b1ba2e8236ecc8","schema_version":"1.0","event_id":"sha256:cba1cbf450c75174da8d14a84e753cfa700d7c7eb71d743973b1ba2e8236ecc8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4WGSR7WDJYDYNB5N4DNMCUWIYK/bundle.json","state_url":"https://pith.science/pith/4WGSR7WDJYDYNB5N4DNMCUWIYK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4WGSR7WDJYDYNB5N4DNMCUWIYK/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":"2026-07-07T09:42:22Z","links":{"resolver":"https://pith.science/pith/4WGSR7WDJYDYNB5N4DNMCUWIYK","bundle":"https://pith.science/pith/4WGSR7WDJYDYNB5N4DNMCUWIYK/bundle.json","state":"https://pith.science/pith/4WGSR7WDJYDYNB5N4DNMCUWIYK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4WGSR7WDJYDYNB5N4DNMCUWIYK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:4WGSR7WDJYDYNB5N4DNMCUWIYK","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":"aef1c79d7d60ebeee027e9b421f51915966a9a756deae324c8ec3d75c01385e5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T17:00:01Z","title_canon_sha256":"13d3cb37a204141cd0e60eece73b5a71e887904e9983dc1e0fb1026c87ca6da3"},"schema_version":"1.0","source":{"id":"2011.09402","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.09402","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"arxiv_version","alias_value":"2011.09402v1","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.09402","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_12","alias_value":"4WGSR7WDJYDY","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_16","alias_value":"4WGSR7WDJYDYNB5N","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_8","alias_value":"4WGSR7WD","created_at":"2026-07-05T01:52:43Z"}],"graph_snapshots":[{"event_id":"sha256:cba1cbf450c75174da8d14a84e753cfa700d7c7eb71d743973b1ba2e8236ecc8","target":"graph","created_at":"2026-07-05T01:52:43Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2011.09402/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For integers $2 \\leq t \\leq k$, we consider a collection of $k$ set families $\\mathcal{A}_j: 1 \\leq j \\leq k$ where $\\mathcal{A}_j = \\{ A_{j,i} \\subseteq [n] : 1 \\leq i \\leq m \\}$ and $|A_{1, i_1} \\cap \\cdots \\cap A_{k,i_k}|$ is even if and only if at least $t$ of the $i_j$ are distinct. In this paper, we prove that $m =O(n^{ 1/ \\lfloor k/2 \\rfloor})$ when $t=k$ and $m = O( n^{1/(t-1)})$ when $2t-2 \\leq k$ and prove that both of these bounds are best possible. Specializing to the case where $\\mathcal{A} = \\mathcal{A}_1 = \\cdots = \\mathcal{A}_k$, we recover a variation of the classical oddtown ","authors_text":"Jacques Verstra\\\"ete, Jason O'Neill","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T17:00:01Z","title":"A note on $k$-wise oddtown problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.09402","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:c31d72f52987cb42b56c7b5084b8cae79b9cc0d55cb88dbccad1b7048b985275","target":"record","created_at":"2026-07-05T01:52:43Z","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":"aef1c79d7d60ebeee027e9b421f51915966a9a756deae324c8ec3d75c01385e5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T17:00:01Z","title_canon_sha256":"13d3cb37a204141cd0e60eece73b5a71e887904e9983dc1e0fb1026c87ca6da3"},"schema_version":"1.0","source":{"id":"2011.09402","kind":"arxiv","version":1}},"canonical_sha256":"e58d28fec34e078687ade0dac152c8c29b16e1e1ed051d668e7926d5472269b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e58d28fec34e078687ade0dac152c8c29b16e1e1ed051d668e7926d5472269b8","first_computed_at":"2026-07-05T01:52:43.654732Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:52:43.654732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LMpWojSqQAcKdbAu3R2gOc2RjKznDPBD+NHYBS6oMdQ1j8PdDtHhUT7fRgAaUNCiYKHHCB5FNIS8OSZwMuVJAg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:52:43.655135Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.09402","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c31d72f52987cb42b56c7b5084b8cae79b9cc0d55cb88dbccad1b7048b985275","sha256:cba1cbf450c75174da8d14a84e753cfa700d7c7eb71d743973b1ba2e8236ecc8"],"state_sha256":"4d1e80d795d0c83ff24733a076abfbe8866cde545e7f01391eeb4a551de59459"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/SabPdQNUqQCmqO3Ax2rM9wHyHiyMLBSQBemQjShxJwRnWGJfKvujheTKTOj9KsR+8CgY7Po0CalUPwAjXnSDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T09:42:22.954343Z","bundle_sha256":"c93dd584ddc627e2acfcbb2b0e7af391be0cba5b5690894bd4b79dc4208e682b"}}