On Monochromatic Pairs of Nondecreasing Diameters

Abstract

Let $n$, $m$, $r$, $t$ be positive integers and $\Delta:[n]\to[r]$. We say $\Delta$ is $(m,r,t)$-permissible if there exist $t$ disjoint $m$-sets $B_1,\dots,B_t$ contained in $[n]$ for which

1. $|\Delta(B_i)|=1$ for each $i=1,2,\dots,t$,
2. $\max(B_i) < \min(B_{i+1})$ for each $i=1,2,\dots,t-1$, and
3. $\max(B_i)-\min(B_i) \leq \max(B_{i+1})-\max(B_{i+1})$ for each $i=1,2,\dots,t-1$.

Let $f(m,r,t)$ be the smallest such $n$ so that all colorings $\Delta$ are $(m,r,t)$-permissible. In this paper, we show that $f(2,2,t)=5t-4$.