New Bounds on Diffsequences

Abstract

For a set of positive integers $D$, a $k$-term $D$-diffsequence is a sequence of positive integers $a_1<a_2<\cdots<a_k$ such that $a_i-a_{i-1}\in D$ for $i=2,3,\cdots,k$. For $k\in\mathbb{Z}^+$ and $D\subset \mathbb{Z}^+$, we define $Δ(D,k)$, if it exists, to be the smallest integer $n$ such that every $2$-coloring of $\{1,2,\cdots,n\}$ contains a monochromatic $D$-diffsequence of length $k$. We improve the lower bound on $Δ(D,k)$ where $D=\{2^i\mid i\in\mathbb{Z}_{\geq{0}}\}$, proving a conjecture of Chokshi, Clifton, Landman, and Sawin. We also determine all sets of the form $D=\{d_1,d_2,\dots\}$ with $d_i\mid d_{i+1}$ for which $Δ(D,k)$ exists.