15:{"metadata":[["$","title","0",{"children":"On Three Sets with Nondecreasing Diameter"}],["$","meta","1",{"name":"description","content":"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, we show that $f(m,m,m;2)=8m-5+\\lfloor\\frac{2m-2}{3}\\rfloor+\\delta$ for $m\\geq 2$, where $\\delta=1$ if $m\\in \\{2,5\\}$ and $\\delta=0$ otherwise."}],["$","meta","2",{"name":"author","content":"Daniel Bernstein"}],["$","meta","3",{"name":"author","content":"David J. Grynkiewicz"}],["$","meta","4",{"name":"author","content":"Carl R. 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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, we show that $f(m,m,m;2)=8m-5+\\lfloor\\frac{2m-2}{3}\\rfloor+\\delta$ for $m\\geq 2$, where $\\delta=1$ if $m\\in \\{2,5\\}$ and $\\delta=0$ otherwise."}],["$","meta","16",{"property":"og:url","content":"https://synthical.com/article/On-Three-Sets-with-Nondecreasing-Diameter-1b65a984-43bc-4147-bb3c-60a80da89ab1"}],["$","meta","17",{"property":"og:site_name","content":"Synthical"}],["$","meta","18",{"property":"og:locale","content":"en-US"}],["$","meta","19",{"property":"og:image","content":"https://synthical.com/preview/article/1b65a984-43bc-4147-bb3c-60a80da89ab1"}],["$","meta","20",{"property":"og:image:alt","content":"On Three Sets with Nondecreasing Diameter"}],["$","meta","21",{"property":"og:image:width","content":"1200"}],["$","meta","22",{"property":"og:image:height","content":"630"}],["$","meta","23",{"property":"og:type","content":"article"}],["$","meta","24",{"property":"article:published_time","content":"2014-07-18T21:38:58.000Z"}],["$","meta","25",{"property":"article:modified_time","content":"2014-07-18T21:38:58.000Z"}],["$","meta","26",{"property":"article:author:name","content":"Daniel Bernstein"}],["$","meta","27",{"property":"article:author:name","content":"David J. 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Improving upon upper and lower bounds of Bialostocki, Erd\\H os and Lefmann, we show that $f(m,m,m;2)=8m-5+\\lfloor\\frac{2m-2}{3}\\rfloor+\\delta$ for $m\\geq 2$, where $\\delta=1$ if $m\\in \\{2,5\\}$ and $\\delta=0$ otherwise."}],["$","meta","40",{"name":"twitter:image","content":"https://synthical.com/preview/article/1b65a984-43bc-4147-bb3c-60a80da89ab1"}],["$","meta","41",{"name":"twitter:image:alt","content":"On Three Sets with Nondecreasing Diameter"}],["$","meta","42",{"name":"twitter:image:width","content":"1200"}],["$","meta","43",{"name":"twitter:image:height","content":"630"}],["$","link","44",{"rel":"icon","href":"/favicon-32x32.png"}],["$","link","45",{"rel":"apple-touch-icon","href":"/apple-touch-icon.png"}]],"error":null,"digest":"$undefined"}