{"paper":{"title":"Chromatic Vertex Folkman Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Meilian Liang, Stanis{\\l}aw Radziszowski, Xiaodong Xu","submitted_at":"2016-12-24T04:03:30Z","abstract_excerpt":"For graph $G$ and integers $a_1 \\ge \\cdots \\ge a_r \\ge 2$, we write $G \\rightarrow (a_1 ,\\cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \\in \\{1, \\cdots, r\\}$. The vertex Folkman number $F_v(a_1 ,\\cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G \\rightarrow (a_1 ,\\cdots ,a_r)^v$. It is well known that if $G \\rightarrow (a_1 ,\\cdots ,a_r)^v$ then $\\chi(G) \\geq m$, where $m = 1+ \\sum_{i=1}^r (a_i - 1)$. In this paper we study"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08136","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":""},"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"}