{"paper":{"title":"The edge Folkman number $F_e(3, 3; 4)$ is greater than 19","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Bikov, Nedyalko Nenov","submitted_at":"2016-09-12T16:32:57Z","abstract_excerpt":"The set of the graphs which do not contain the complete graph on $q$ vertices $K_q$ and have the property that in every coloring of their edges in two colors there exist a monochromatic triangle is denoted by $\\mathcal{H}_e(3, 3; q)$. The edge Folkman numbers $F_e(3, 3; q) = \\min\\{|V(G)| : G \\in \\mathcal{H}_e(3, 3; q)\\}$ are considered. Folkman proved in 1970 that $F_e(3, 3; q)$ exists if and only if $q \\geq 4$. From the Ramsey number $R(3, 3) = 6$ it becomes clear that $F_e(3, 3; q) = 6$ if $q \\geq 7$. It is also known that $F_e(3, 3; 6) = 8$ and $F_e(3, 3; 5) = 15$. The upper bounds on the n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.03468","kind":"arxiv","version":1},"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"}