{"paper":{"title":"An exponential-type upper bound for Folkman numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej Ruci\\'nski, Mathias Schacht, Vojt\\v{e}ch R\\\"odl","submitted_at":"2016-03-01T22:59:04Z","abstract_excerpt":"For given integers $k$ and $r$, the Folkman number $f(k;r)$ is the smallest number of vertices in a graph $G$ which contains no clique on $k+1$ vertices, yet for every partition of its edges into $r$ parts, some part contains a clique of order $k$. The existence (finiteness) of Folkman numbers was established by Folkman (1970) for $r=2$ and by Ne\\v{s}et\\v{r}il and R\\\"odl (1976) for arbitrary $r$, but these proofs led to very weak upper bounds on $f(k;r)$.\n  Recently, Conlon and Gowers and independently the authors obtained a doubly exponential bound on $f(k;2)$. Here, we establish a further im"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00521","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"}