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The vertex Folkman numbers $$F_v(a_1, ..., a_s; m - 1) = \\min\\{\\vert V(G)\\vert : G \\overset{v}{\\rightarrow} (a_1, ..., a_s) \\mbox{ and } K_{m - 1} \\not\\subseteq G\\}.$$ are considered, where $m = \\sum_{i = 1}^{s}(a_i - 1) + 1$. We know the exact values of all the numbers $F_v(a_1, ..., a_s; m - 1)$ when $\\max\\{a_1, ..., a_s\\} \\leq 6$ and also the number $F_v(2, 2, 7; 8"},"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":"1711.01535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-05T05:38:44Z","cross_cats_sorted":[],"title_canon_sha256":"ec47be59cb70ce3fc63bf6e24905b2189dbc79e7cb43bcccc1194beae4b50994","abstract_canon_sha256":"eb3bbe62ac42d64b1a9cc90bb30f00de2716ece8583305124d48b58cb41fd1c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:07.498166Z","signature_b64":"laHMov8Uuz/fpk7zuiRbEqDlHN8CLQskD9p/SYlJQDlwb5IfcZKJyZaX2FcRT093puZTS0LI5gJUcWK+rC2rCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8640567719b04c1e1764970c0bafa043689a251919be1d699683215bf6c21cc","last_reissued_at":"2026-05-17T23:50:07.497582Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:07.497582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower bounding the Folkman numbers $F_v(a_1, ..., a_s; m - 1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Bikov, Nedyalko Nenov","submitted_at":"2017-11-05T05:38:44Z","abstract_excerpt":"For a graph $G$ the expression $G \\overset{v}{\\rightarrow} (a_1, ..., a_s)$ means that for every $s$-coloring of the vertices of $G$ there exists $i \\in \\{1, ..., s\\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers $$F_v(a_1, ..., a_s; m - 1) = \\min\\{\\vert V(G)\\vert : G \\overset{v}{\\rightarrow} (a_1, ..., a_s) \\mbox{ and } K_{m - 1} \\not\\subseteq G\\}.$$ are considered, where $m = \\sum_{i = 1}^{s}(a_i - 1) + 1$. We know the exact values of all the numbers $F_v(a_1, ..., a_s; m - 1)$ when $\\max\\{a_1, ..., a_s\\} \\leq 6$ and also the number $F_v(2, 2, 7; 8"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.01535","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1711.01535","created_at":"2026-05-17T23:50:07.497674+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.01535v1","created_at":"2026-05-17T23:50:07.497674+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.01535","created_at":"2026-05-17T23:50:07.497674+00:00"},{"alias_kind":"pith_short_12","alias_value":"VBSAKZ3RTMCM","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"VBSAKZ3RTMCMDYLW","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"VBSAKZ3R","created_at":"2026-05-18T12:31:49.984773+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/VBSAKZ3RTMCMDYLWJFYMBOX2AQ","json":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ.json","graph_json":"https://pith.science/api/pith-number/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/graph.json","events_json":"https://pith.science/api/pith-number/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/events.json","paper":"https://pith.science/paper/VBSAKZ3R"},"agent_actions":{"view_html":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ","download_json":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ.json","view_paper":"https://pith.science/paper/VBSAKZ3R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.01535&json=true","fetch_graph":"https://pith.science/api/pith-number/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/graph.json","fetch_events":"https://pith.science/api/pith-number/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/action/storage_attestation","attest_author":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/action/author_attestation","sign_citation":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/action/citation_signature","submit_replication":"https://pith.science/pith/VBSAKZ3RTMCMDYLWJFYMBOX2AQ/action/replication_record"}},"created_at":"2026-05-17T23:50:07.497674+00:00","updated_at":"2026-05-17T23:50:07.497674+00:00"}