On Three Sets with Nondecreasing Diameter

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 $\mathrm{\Delta }:[1,N]\to \{0,1\}$ 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 $diam(B_1)\le diam(B_2)\le diam(B_3)$. 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 $m\ge 2$, where $\delta =1$ if $m\in \{2,5\}$ and $\delta =0$ otherwise.