{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:KHMZXFZBGFKKRWO65FWWQ2AUSU","short_pith_number":"pith:KHMZXFZB","schema_version":"1.0","canonical_sha256":"51d99b97213154a8d9dee96d686814952641d8e5f3be90ef3e37d39cd443e461","source":{"kind":"arxiv","id":"1502.00445","version":3},"attestation_state":"computed","paper":{"title":"Random-Player Maker-Breaker games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Gal Kronenberg, Michael Krivelevich","submitted_at":"2015-02-02T11:57:07Z","abstract_excerpt":"In a $(1:b)$ Maker-Breaker game, a primary question is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erd\\H{o}s conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, \"Erd\\H{o}s Paradigm\" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly.\n  A random-player Maker-Breaker game is a two"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1502.00445","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-02T11:57:07Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"b9a59a541ad9eff2b4a5b55d6d8ba00161ca9bc3cb1b0fa1f99876d6bdfed9aa","abstract_canon_sha256":"f11c65cf34794ccf6a7939c9bc7f884216265c4d1d0a0acf6c202913ec761459"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:12.413039Z","signature_b64":"MEqEUkSlwpArWPgRK3OKQOd6CFVJl1vRptjSa+mtA/Pht/NzlQ1IxH/2QO2rrf2zVYI5OgF2kxPELoTKXrlSCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51d99b97213154a8d9dee96d686814952641d8e5f3be90ef3e37d39cd443e461","last_reissued_at":"2026-05-18T01:19:12.412371Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:12.412371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random-Player Maker-Breaker games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Gal Kronenberg, Michael Krivelevich","submitted_at":"2015-02-02T11:57:07Z","abstract_excerpt":"In a $(1:b)$ Maker-Breaker game, a primary question is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erd\\H{o}s conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, \"Erd\\H{o}s Paradigm\" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly.\n  A random-player Maker-Breaker game is a two"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00445","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1502.00445","created_at":"2026-05-18T01:19:12.412477+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.00445v3","created_at":"2026-05-18T01:19:12.412477+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00445","created_at":"2026-05-18T01:19:12.412477+00:00"},{"alias_kind":"pith_short_12","alias_value":"KHMZXFZBGFKK","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"KHMZXFZBGFKKRWO6","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"KHMZXFZB","created_at":"2026-05-18T12:29:27.538025+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU","json":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU.json","graph_json":"https://pith.science/api/pith-number/KHMZXFZBGFKKRWO65FWWQ2AUSU/graph.json","events_json":"https://pith.science/api/pith-number/KHMZXFZBGFKKRWO65FWWQ2AUSU/events.json","paper":"https://pith.science/paper/KHMZXFZB"},"agent_actions":{"view_html":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU","download_json":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU.json","view_paper":"https://pith.science/paper/KHMZXFZB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.00445&json=true","fetch_graph":"https://pith.science/api/pith-number/KHMZXFZBGFKKRWO65FWWQ2AUSU/graph.json","fetch_events":"https://pith.science/api/pith-number/KHMZXFZBGFKKRWO65FWWQ2AUSU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU/action/storage_attestation","attest_author":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU/action/author_attestation","sign_citation":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU/action/citation_signature","submit_replication":"https://pith.science/pith/KHMZXFZBGFKKRWO65FWWQ2AUSU/action/replication_record"}},"created_at":"2026-05-18T01:19:12.412477+00:00","updated_at":"2026-05-18T01:19:12.412477+00:00"}