{"paper":{"title":"On the vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1)$ when $\\max\\{a_1, ..., a_s\\} = 6$ or $7$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Bikov, Nedyalko Nenov","submitted_at":"2015-12-07T14:09:14Z","abstract_excerpt":"Let $G$ be a graph and $a_1, ..., a_s$ be positive integers. Then $G \\overset{v}{\\rightarrow} (a_1, ..., a_s)$ means that for every coloring of the vertices of $G$ in $s$ colors there exists $i \\in \\{1, ..., s\\}$, such that there is a monochromatic $a_i$-clique of color $i$. The vertex Folkman number $F_v(a_1, ..., a_s; q)$ is defined by the equality: $$ F_v(a_1, ..., a_s; q) = \\min\\{|V(G)| : G \\overset{v}{\\rightarrow} (a_1, ..., a_s) \\mbox{ and } K_q \\not\\subseteq G\\}. $$ Let $m = \\sum\\limits_{i=1}^s (a_i - 1) + 1$. It is easy to see that $F_v(a_1, ..., a_s; q) = m$ if $q \\geq m + 1$. In [11]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02051","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"}