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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 $m = \\sum_{i = 1}^{s}(a_i - 1) + 1$.\n  With the help of computer we show that $F_v(2, 2, 5; 6) = 16$ and then we prove $$F_v(a_1, ..., a_s; m - 1) = m + 9,$$ if $\\max\\{a_1, ..., a_s"},"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":"1503.08444","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-29T15:27:02Z","cross_cats_sorted":[],"title_canon_sha256":"e83d4d3e5de756dd9b6aa49e2d510d689cbbb601bd38847cd8a49c36ad16c89a","abstract_canon_sha256":"c51588d6d4d3868a32f35f46241715596f28769bbb899ad705689c1b1036dc1d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:07.737135Z","signature_b64":"JANTiMFZFGEzsCQbGJ9mPRz4bpVM3xlEMjC5ljII+R4sFQPOuw7kx+bi4mQylhAsISTAhkXj6Zyk7lHwlGd0Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5448eaa2279c5c818661bee682d6ac1fab76049d784eab301e2050ef95fe19c4","last_reissued_at":"2026-05-17T23:50:07.736373Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:07.736373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1) = m + 9$, if $\\max\\{a_1, ..., a_s\\} = 5$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Bikov, Nedyalko Nenov","submitted_at":"2015-03-29T15:27:02Z","abstract_excerpt":"For a graph $G$ the expression $G \\overset{v}{\\rightarrow} (a_1, ..., a_s)$ means that for any $s$-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 $m = \\sum_{i = 1}^{s}(a_i - 1) + 1$.\n  With the help of computer we show that $F_v(2, 2, 5; 6) = 16$ and then we prove $$F_v(a_1, ..., a_s; m - 1) = m + 9,$$ if $\\max\\{a_1, ..., a_s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08444","kind":"arxiv","version":2},"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":"1503.08444","created_at":"2026-05-17T23:50:07.736498+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.08444v2","created_at":"2026-05-17T23:50:07.736498+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08444","created_at":"2026-05-17T23:50:07.736498+00:00"},{"alias_kind":"pith_short_12","alias_value":"KREOVIRHTROI","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"KREOVIRHTROIDBTB","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"KREOVIRH","created_at":"2026-05-18T12:29:29.992203+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/KREOVIRHTROIDBTBX3TIFVVMD6","json":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6.json","graph_json":"https://pith.science/api/pith-number/KREOVIRHTROIDBTBX3TIFVVMD6/graph.json","events_json":"https://pith.science/api/pith-number/KREOVIRHTROIDBTBX3TIFVVMD6/events.json","paper":"https://pith.science/paper/KREOVIRH"},"agent_actions":{"view_html":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6","download_json":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6.json","view_paper":"https://pith.science/paper/KREOVIRH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.08444&json=true","fetch_graph":"https://pith.science/api/pith-number/KREOVIRHTROIDBTBX3TIFVVMD6/graph.json","fetch_events":"https://pith.science/api/pith-number/KREOVIRHTROIDBTBX3TIFVVMD6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6/action/storage_attestation","attest_author":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6/action/author_attestation","sign_citation":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6/action/citation_signature","submit_replication":"https://pith.science/pith/KREOVIRHTROIDBTBX3TIFVVMD6/action/replication_record"}},"created_at":"2026-05-17T23:50:07.736498+00:00","updated_at":"2026-05-17T23:50:07.736498+00:00"}