{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:LWY535OP335ZG3UOVAVPNOLJVQ","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":"1175214dd0d2ebc9321f51a46f8b89ee26d366aeb1e1f38a1c61e02c21c7ed63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-13T10:40:39Z","title_canon_sha256":"7fc322d2a76bbdd92e510f9076530dcaa40491c991a2036b3f4b151517d8a9a7"},"schema_version":"1.0","source":{"id":"1604.03716","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.03716","created_at":"2026-05-17T23:50:07Z"},{"alias_kind":"arxiv_version","alias_value":"1604.03716v1","created_at":"2026-05-17T23:50:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.03716","created_at":"2026-05-17T23:50:07Z"},{"alias_kind":"pith_short_12","alias_value":"LWY535OP335Z","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LWY535OP335ZG3UO","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LWY535OP","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:ff2f209049e0924fb3672accade03e79a25cb1f445bf790beb9a86a61c9f7d8f","target":"graph","created_at":"2026-05-17T23:50:07Z","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 say that $G$ is a $(3, 3)$-Ramsey graph if every $2$-coloring of the edges of $G$ forces a monochromatic triangle. The $(3, 3)$-Ramsey graph $G$ is minimal if $G$ does not contain a proper $(3, 3)$-Ramsey subgraph. In this work we find all minimal $(3, 3)$-Ramsey graphs with up to 13 vertices with the help of a computer, and we obtain some new results for these graphs. We also obtain new upper bounds on the independence number and new lower bounds on the minimum degree of arbitrary $(3, 3)$-Ramsey graphs.","authors_text":"Aleksandar Bikov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-13T10:40:39Z","title":"Small minimal $(3, 3)$-Ramsey graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.03716","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":"sha256:096150ad8fca1d563b78e6a29e0cc70c907bd678469b4b040c481690ff80f250","target":"record","created_at":"2026-05-17T23:50:07Z","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":"1175214dd0d2ebc9321f51a46f8b89ee26d366aeb1e1f38a1c61e02c21c7ed63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-13T10:40:39Z","title_canon_sha256":"7fc322d2a76bbdd92e510f9076530dcaa40491c991a2036b3f4b151517d8a9a7"},"schema_version":"1.0","source":{"id":"1604.03716","kind":"arxiv","version":1}},"canonical_sha256":"5db1ddf5cfdefb936e8ea82af6b969ac3e7e829652103041c7df50328af4fb75","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5db1ddf5cfdefb936e8ea82af6b969ac3e7e829652103041c7df50328af4fb75","first_computed_at":"2026-05-17T23:50:07.670532Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:07.670532Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"15Pmpp1Q8w74nuailLdGBYTzb2eNicmLvldhujRFBnQWcCIUa4v2QG96FbYhfxMe4Ux3swBt2sQUr3ky3rgpCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:07.671171Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.03716","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:096150ad8fca1d563b78e6a29e0cc70c907bd678469b4b040c481690ff80f250","sha256:ff2f209049e0924fb3672accade03e79a25cb1f445bf790beb9a86a61c9f7d8f"],"state_sha256":"868fdfe63c13d8ad13361079485431326fb4a0a254f2d6d6edccd824c273f0cf"}