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The vertex Folkman numbers are defined as $F_v(a_1,\\dots,a_r;H) = \\min\\{|V(G)| : G$ is $H$-free and $G \\rightarrow (a_1,\\dots,a_r)^v\\}$, where $H$ is a graph. Such vertex Folkman numbers have been extensively studied for $H=K_s$ with $s>\\max\\{a_i\\}_{1\\le i \\le r}$. If $a_i=a$ for all $i$, then we use notation $F_v(a^r;H)=F_v(a_1,\\dots,a_r;H)$.\n  Let $J_k$ be the co"},"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":"2110.03121","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-10-07T00:33:25Z","cross_cats_sorted":[],"title_canon_sha256":"ab71e93f326d805e2cf8da500f51f92aec235b48040a662eda0f8426b19f8edd","abstract_canon_sha256":"05be6b8944f874ce00b62e843588b9c06204c7bfd8d16851edc5eff36d5a8b27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:23:29.061895Z","signature_b64":"2R29M3KkhDj2iX4jfyif4FVz9r3SdDVP8DXVLXDg2iMB+gWq9KXU93rHsf95XGfosEQ4OYYBumoJmVkcHr9/Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c410dbf78e17229ef913c77652bbf17220cd7be71f438807e3f1898c83c20aa","last_reissued_at":"2026-07-05T05:23:29.061409Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:23:29.061409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Some Generalized Vertex Folkman Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David E. Narv\\'aez, Stanis{\\l}aw Radziszowski, Xiaodong Xu, Yu Jiang, Zohair Raza Hassan","submitted_at":"2021-10-07T00:33:25Z","abstract_excerpt":"For a graph $G$ and integers $a_i\\ge 1$, the expression $G \\rightarrow (a_1,\\dots,a_r)^v$ means that for any $r$-coloring of the vertices of $G$ there exists a monochromatic $a_i$-clique in $G$ for some color $i \\in \\{1,\\cdots,r\\}$. The vertex Folkman numbers are defined as $F_v(a_1,\\dots,a_r;H) = \\min\\{|V(G)| : G$ is $H$-free and $G \\rightarrow (a_1,\\dots,a_r)^v\\}$, where $H$ is a graph. Such vertex Folkman numbers have been extensively studied for $H=K_s$ with $s>\\max\\{a_i\\}_{1\\le i \\le r}$. 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