{"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 $s>\\max\\{a_i\\}_{1\\le i \\le r}$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.03121","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.03121/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}