New Bounds on Diffsequences

TL;DR

This paper establishes new bounds on the minimal length of sequences needed to guarantee monochromatic diffsequences in 2-colorings, specifically improving bounds for exponential difference sets and characterizing when such bounds exist.

Contribution

It improves the lower bounds on elta(D,k) for exponential difference sets and characterizes all difference sets with divisibility properties for which elta(D,k) exists.

Findings

Improved lower bounds on elta(D,k) for D=rac{1}{2}^i

Resolved a conjecture by Chokshi et al.

Characterized all divisibility-structured sets D for which elta(D,k) exists.

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 $\Delta(D,k)$, 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 $\Delta(D,k)$ 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 $\Delta(D,k)$ exists.