{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:PTGA4FDUOMYZRV6T5PVOA2FPZ2","short_pith_number":"pith:PTGA4FDU","canonical_record":{"source":{"id":"1108.0710","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-02T23:14:22Z","cross_cats_sorted":[],"title_canon_sha256":"f6fca54788e4eac066b25a937c38f0c1201ac6a98d2d33f9699ed07088932d44","abstract_canon_sha256":"cb47a522f75704590c274cc1762209468a8dbb2ae54bf2c4bb285bcc3b7f7241"},"schema_version":"1.0"},"canonical_sha256":"7ccc0e1474733198d7d3ebeae068afcea846f8ec5ecb227ea66e8351af62d3a5","source":{"kind":"arxiv","id":"1108.0710","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0710","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0710v1","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0710","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"pith_short_12","alias_value":"PTGA4FDUOMYZ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PTGA4FDUOMYZRV6T","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PTGA4FDU","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:PTGA4FDUOMYZRV6T5PVOA2FPZ2","target":"record","payload":{"canonical_record":{"source":{"id":"1108.0710","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-02T23:14:22Z","cross_cats_sorted":[],"title_canon_sha256":"f6fca54788e4eac066b25a937c38f0c1201ac6a98d2d33f9699ed07088932d44","abstract_canon_sha256":"cb47a522f75704590c274cc1762209468a8dbb2ae54bf2c4bb285bcc3b7f7241"},"schema_version":"1.0"},"canonical_sha256":"7ccc0e1474733198d7d3ebeae068afcea846f8ec5ecb227ea66e8351af62d3a5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:53.099977Z","signature_b64":"UFoZqeemXD5jrksYNqztC2h2lCUPCdk0KjG2OH1509wulUHRxJ7Iz7iVzlfBZK85oiKdSIiLAIx0ye86+qUpAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ccc0e1474733198d7d3ebeae068afcea846f8ec5ecb227ea66e8351af62d3a5","last_reissued_at":"2026-05-18T01:35:53.099359Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:53.099359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.0710","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":"2026-05-18T01:35:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9370M5zrnYpD0gPSSAnkm9HFKNUeRvxfA5RLjgS7uOpUaHi3z2nQ8PrwYc18gIBnd4IELN3L6YvUgd7yrj4mBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T11:50:28.621518Z"},"content_sha256":"6a5510ea1ee0d241cdec1bf6b9c5daf699e7fb868efa4e861d452d6826b06f63","schema_version":"1.0","event_id":"sha256:6a5510ea1ee0d241cdec1bf6b9c5daf699e7fb868efa4e861d452d6826b06f63"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:PTGA4FDUOMYZRV6T5PVOA2FPZ2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Chain-making games in grid-like posets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel W. Cranston, Douglas B. West, Gregory J. Puleo, Kevin G. Milans, William B. Kinnersley","submitted_at":"2011-08-02T23:14:22Z","abstract_excerpt":"We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset. In a product of chains, the maximum size of a chain that Maker can guarantee building is $k-\\lfloor r/2\\rfloor$, where $k$ is the maximum size of a chain in the product, and $r$ is the maximum size of a factor chain. We also study a variant in which Maker must follow the chain in order, called the {\\it Walker-Blocker game}. In the poset consisting of the bottom $k$ levels of the product of $d$ arbitrarily long chains, Walker can guarantee a chain that hits all levels if $d\\ge14$; this result uses a solution t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0710","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":"2026-05-18T01:35:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oCCa4RvaOC7yZeEmplsv1tX7yYp6GleorXydWE+4kJTDNl1qVrEhh9lWZjYB+VBtUEdadK4pebGCP6TRoxvdAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T11:50:28.621866Z"},"content_sha256":"01e7b52cec63a4245d1125a8bb78bffc1c028e4bedfd8f2f2fb0390481dac48c","schema_version":"1.0","event_id":"sha256:01e7b52cec63a4245d1125a8bb78bffc1c028e4bedfd8f2f2fb0390481dac48c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PTGA4FDUOMYZRV6T5PVOA2FPZ2/bundle.json","state_url":"https://pith.science/pith/PTGA4FDUOMYZRV6T5PVOA2FPZ2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PTGA4FDUOMYZRV6T5PVOA2FPZ2/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":"2026-06-21T11:50:28Z","links":{"resolver":"https://pith.science/pith/PTGA4FDUOMYZRV6T5PVOA2FPZ2","bundle":"https://pith.science/pith/PTGA4FDUOMYZRV6T5PVOA2FPZ2/bundle.json","state":"https://pith.science/pith/PTGA4FDUOMYZRV6T5PVOA2FPZ2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PTGA4FDUOMYZRV6T5PVOA2FPZ2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:PTGA4FDUOMYZRV6T5PVOA2FPZ2","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":"cb47a522f75704590c274cc1762209468a8dbb2ae54bf2c4bb285bcc3b7f7241","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-02T23:14:22Z","title_canon_sha256":"f6fca54788e4eac066b25a937c38f0c1201ac6a98d2d33f9699ed07088932d44"},"schema_version":"1.0","source":{"id":"1108.0710","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0710","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0710v1","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0710","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"pith_short_12","alias_value":"PTGA4FDUOMYZ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PTGA4FDUOMYZRV6T","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PTGA4FDU","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:01e7b52cec63a4245d1125a8bb78bffc1c028e4bedfd8f2f2fb0390481dac48c","target":"graph","created_at":"2026-05-18T01:35:53Z","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":"We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset. In a product of chains, the maximum size of a chain that Maker can guarantee building is $k-\\lfloor r/2\\rfloor$, where $k$ is the maximum size of a chain in the product, and $r$ is the maximum size of a factor chain. We also study a variant in which Maker must follow the chain in order, called the {\\it Walker-Blocker game}. In the poset consisting of the bottom $k$ levels of the product of $d$ arbitrarily long chains, Walker can guarantee a chain that hits all levels if $d\\ge14$; this result uses a solution t","authors_text":"Daniel W. Cranston, Douglas B. West, Gregory J. Puleo, Kevin G. Milans, William B. Kinnersley","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-02T23:14:22Z","title":"Chain-making games in grid-like posets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0710","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":"sha256:6a5510ea1ee0d241cdec1bf6b9c5daf699e7fb868efa4e861d452d6826b06f63","target":"record","created_at":"2026-05-18T01:35:53Z","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":"cb47a522f75704590c274cc1762209468a8dbb2ae54bf2c4bb285bcc3b7f7241","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-02T23:14:22Z","title_canon_sha256":"f6fca54788e4eac066b25a937c38f0c1201ac6a98d2d33f9699ed07088932d44"},"schema_version":"1.0","source":{"id":"1108.0710","kind":"arxiv","version":1}},"canonical_sha256":"7ccc0e1474733198d7d3ebeae068afcea846f8ec5ecb227ea66e8351af62d3a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ccc0e1474733198d7d3ebeae068afcea846f8ec5ecb227ea66e8351af62d3a5","first_computed_at":"2026-05-18T01:35:53.099359Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:53.099359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UFoZqeemXD5jrksYNqztC2h2lCUPCdk0KjG2OH1509wulUHRxJ7Iz7iVzlfBZK85oiKdSIiLAIx0ye86+qUpAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:53.099977Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.0710","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6a5510ea1ee0d241cdec1bf6b9c5daf699e7fb868efa4e861d452d6826b06f63","sha256:01e7b52cec63a4245d1125a8bb78bffc1c028e4bedfd8f2f2fb0390481dac48c"],"state_sha256":"34b6c7a578cce62422591d66ea911dfacd4e36d84dfa750ec29b94f7f8f91476"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XuxJ0VfQDP7G09NcIl1G+8SSuLYFxnA0XTN1UajfS5KypUhZyWEyuJo2Nd5zxMwRuFwp9VAB5QWs5jVdOfauBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T11:50:28.623829Z","bundle_sha256":"6c8a77c7b5c18de95d2de668bd7edb1f92373e2ce53bbfaeebad6029a48dd5c9"}}