{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:MWMS7XIH44JNURNBZR7GSGIMIG","short_pith_number":"pith:MWMS7XIH","canonical_record":{"source":{"id":"2606.24198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T06:36:28Z","cross_cats_sorted":[],"title_canon_sha256":"1de2ac6b5bf15dae3b0c96f87bf3e733f1359ada2b2d3eebc16530f436c359a2","abstract_canon_sha256":"accc9987b57deca2eb70dba83860d0de9a28e23bafc353e3d4bdffe53b6602a8"},"schema_version":"1.0"},"canonical_sha256":"65992fdd07e712da45a1cc7e69190c419816827bdb11e7a08cfb8ee0331b4e99","source":{"kind":"arxiv","id":"2606.24198","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24198","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24198v1","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24198","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"pith_short_12","alias_value":"MWMS7XIH44JN","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"pith_short_16","alias_value":"MWMS7XIH44JNURNB","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"pith_short_8","alias_value":"MWMS7XIH","created_at":"2026-06-24T01:14:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:MWMS7XIH44JNURNBZR7GSGIMIG","target":"record","payload":{"canonical_record":{"source":{"id":"2606.24198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T06:36:28Z","cross_cats_sorted":[],"title_canon_sha256":"1de2ac6b5bf15dae3b0c96f87bf3e733f1359ada2b2d3eebc16530f436c359a2","abstract_canon_sha256":"accc9987b57deca2eb70dba83860d0de9a28e23bafc353e3d4bdffe53b6602a8"},"schema_version":"1.0"},"canonical_sha256":"65992fdd07e712da45a1cc7e69190c419816827bdb11e7a08cfb8ee0331b4e99","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:14:45.553128Z","signature_b64":"f86zpvILcJS3FvkIucmF5JJwHTXoBaHcKCHb36VI9EWWUGWSL/wiX8CJq0dLMTpsS61cVGoyoQTYdFvuIL4NBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65992fdd07e712da45a1cc7e69190c419816827bdb11e7a08cfb8ee0331b4e99","last_reissued_at":"2026-06-24T01:14:45.552276Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:14:45.552276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.24198","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":"2026-06-24T01:14:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1kNQF5s+gkgPyMGoVujBVRTpFLwxv1iP4I+ZsdiB1j1nbWY9DJqfJgIjYdwzKMljSmgglJvw1UHTGLL+lPEMBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T01:54:14.393003Z"},"content_sha256":"b9bad200167f55ac2185f723882aa9044e8b1fc964105ebf301ef6bc1ce16704","schema_version":"1.0","event_id":"sha256:b9bad200167f55ac2185f723882aa9044e8b1fc964105ebf301ef6bc1ce16704"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:MWMS7XIH44JNURNBZR7GSGIMIG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"New Tower-Type Lower Bounds for Hypergraph Ramsey Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guanghui Wang, Hanzhi Bai, Longma Du, Ruilong Liu, Xinyu Hu","submitted_at":"2026-06-23T06:36:28Z","abstract_excerpt":"The Ramsey number $r_k(s,m)$ is the smallest $N$ such that any red/blue coloring of the $k$-subsets of $[N]$ contains a red $s$-set or a blue $m$-set. For fixed $k$ and $s$, and for sufficiently large $m$, the tower growth rate is determined by the stepping-up lemma, but for $s=m=k+1$ the available stepping-up lemmas do not apply. Fox asked for estimates of $r_k(k+1,k+1)$. Pudl\\'ak, R\\\"odl, and Wesley gave the first tower-type bound: $r_k(k+1,k+1)\\ge s_3(\\lfloor k/4\\rfloor)\\ge 4\\operatorname{twr}_{\\lfloor k/4\\rfloor-4}(2)$, where $s_3(k)$ is the $3$-color shift number and $\\operatorname{twr}_1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24198","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/2606.24198/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":"2026-06-24T01:14:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YxafVlnzPvfmHt7+oQVwyk8Bz0vVwCCqD3EJCMG3qWP/Gc5yQKR3ZdgsP6X0srncyI5T/Fh5tL7xrMxkDcDWCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T01:54:14.393398Z"},"content_sha256":"e2824a6a01363e5261731c767bbb5a867b9f20cf4237841aec874c3d2f077017","schema_version":"1.0","event_id":"sha256:e2824a6a01363e5261731c767bbb5a867b9f20cf4237841aec874c3d2f077017"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MWMS7XIH44JNURNBZR7GSGIMIG/bundle.json","state_url":"https://pith.science/pith/MWMS7XIH44JNURNBZR7GSGIMIG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MWMS7XIH44JNURNBZR7GSGIMIG/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":"2026-06-30T01:54:14Z","links":{"resolver":"https://pith.science/pith/MWMS7XIH44JNURNBZR7GSGIMIG","bundle":"https://pith.science/pith/MWMS7XIH44JNURNBZR7GSGIMIG/bundle.json","state":"https://pith.science/pith/MWMS7XIH44JNURNBZR7GSGIMIG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MWMS7XIH44JNURNBZR7GSGIMIG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MWMS7XIH44JNURNBZR7GSGIMIG","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":"accc9987b57deca2eb70dba83860d0de9a28e23bafc353e3d4bdffe53b6602a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T06:36:28Z","title_canon_sha256":"1de2ac6b5bf15dae3b0c96f87bf3e733f1359ada2b2d3eebc16530f436c359a2"},"schema_version":"1.0","source":{"id":"2606.24198","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24198","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24198v1","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24198","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"pith_short_12","alias_value":"MWMS7XIH44JN","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"pith_short_16","alias_value":"MWMS7XIH44JNURNB","created_at":"2026-06-24T01:14:45Z"},{"alias_kind":"pith_short_8","alias_value":"MWMS7XIH","created_at":"2026-06-24T01:14:45Z"}],"graph_snapshots":[{"event_id":"sha256:e2824a6a01363e5261731c767bbb5a867b9f20cf4237841aec874c3d2f077017","target":"graph","created_at":"2026-06-24T01:14:45Z","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/2606.24198/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Ramsey number $r_k(s,m)$ is the smallest $N$ such that any red/blue coloring of the $k$-subsets of $[N]$ contains a red $s$-set or a blue $m$-set. For fixed $k$ and $s$, and for sufficiently large $m$, the tower growth rate is determined by the stepping-up lemma, but for $s=m=k+1$ the available stepping-up lemmas do not apply. Fox asked for estimates of $r_k(k+1,k+1)$. Pudl\\'ak, R\\\"odl, and Wesley gave the first tower-type bound: $r_k(k+1,k+1)\\ge s_3(\\lfloor k/4\\rfloor)\\ge 4\\operatorname{twr}_{\\lfloor k/4\\rfloor-4}(2)$, where $s_3(k)$ is the $3$-color shift number and $\\operatorname{twr}_1","authors_text":"Guanghui Wang, Hanzhi Bai, Longma Du, Ruilong Liu, Xinyu Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T06:36:28Z","title":"New Tower-Type Lower Bounds for Hypergraph Ramsey Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24198","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:b9bad200167f55ac2185f723882aa9044e8b1fc964105ebf301ef6bc1ce16704","target":"record","created_at":"2026-06-24T01:14:45Z","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":"accc9987b57deca2eb70dba83860d0de9a28e23bafc353e3d4bdffe53b6602a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T06:36:28Z","title_canon_sha256":"1de2ac6b5bf15dae3b0c96f87bf3e733f1359ada2b2d3eebc16530f436c359a2"},"schema_version":"1.0","source":{"id":"2606.24198","kind":"arxiv","version":1}},"canonical_sha256":"65992fdd07e712da45a1cc7e69190c419816827bdb11e7a08cfb8ee0331b4e99","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"65992fdd07e712da45a1cc7e69190c419816827bdb11e7a08cfb8ee0331b4e99","first_computed_at":"2026-06-24T01:14:45.552276Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-24T01:14:45.552276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f86zpvILcJS3FvkIucmF5JJwHTXoBaHcKCHb36VI9EWWUGWSL/wiX8CJq0dLMTpsS61cVGoyoQTYdFvuIL4NBg==","signature_status":"signed_v1","signed_at":"2026-06-24T01:14:45.553128Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.24198","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9bad200167f55ac2185f723882aa9044e8b1fc964105ebf301ef6bc1ce16704","sha256:e2824a6a01363e5261731c767bbb5a867b9f20cf4237841aec874c3d2f077017"],"state_sha256":"3bef16f373caa711c5aa37a43debf7c14f72316bf8e81dac6b673b1042c10656"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nooq1JqzUdeawWviFeNJ4zf3+RrZm6StHDK2RvvBKqGhnf44Y9qerUj3ukeoiJKWdUVeo2aIOkhLJWx6P2vDCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T01:54:14.395874Z","bundle_sha256":"111be31b3f9ec8fb99454ea3a4c10afddc937e5fcb482d5b28935807539a2423"}}