On Three Sets with Nondecreasing Diameter

Abstract: 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.