{"paper":{"title":"On Three Sets with Nondecreasing Diameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carl R. Yerger, Daniel Bernstein, David J. Grynkiewicz","submitted_at":"2014-07-18T21:38:58Z","abstract_excerpt":"Let $[a,b]$ denote the integers between $a$ and $b$ inclusive and, for a finite subset $X \\subseteq \\mathbb{Z}$, let the diameter of $X$ be equal to $\\max(X)-\\min(X)$. We write $X<_p\\,Y$ provided $\\max(X)<\\min(Y)$. For a positive integer $m$, let $f(m,m,m;2)$ be the least integer $N$ such that any $2$-coloring $\\Delta: [1, N]\\rightarrow \\{0,1\\}$ has three monochromatic $m$-sets $B_1, B_2, B_3 \\subseteq [1,N]$ (not necessarily of the same color) with $B_1<_p\\, B_2 <_p\\, B_3$ and $diam(B_1)\\leq diam(B_2)\\leq diam(B_3)$. Improving upon upper and lower bounds of Bialostocki, Erd\\H os and Lefmann, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5122","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"}