{"paper":{"title":"On generalized Ramsey numbers of Erd\\H{o}s and Rogers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej Dudek, Troy Retter, Vojta R\\\"odl","submitted_at":"2013-09-18T01:56:39Z","abstract_excerpt":"Extending the concept of Ramsey numbers, Erd{\\H o}s and Rogers introduced the following function. For given integers $2\\le s<t$ let $$ f_{s,t}(n)=\\min \\{\\max \\{|W| : W\\subseteq V(G) {and} G[W] {contains no} K_s\\} \\}, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this paper, we show that for every $s\\ge 3$ there exist constants $c_1=c_1(s)$ and $c_2=c_2(s)$ such that $f_{s,s+1}(n) \\le c_1 (\\log n)^{c_2} \\sqrt{n}$. This result is best possible up to a polylogarithmic factor. We also show for all $t-2 \\geq s \\geq 4$, there exists a constant $c_3$ such that $f_{s,t}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4521","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"}