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 $\Delta (D,k)$, 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 $D=\{2^i\mid i\in {\mathbb{Z}}_{\ge 0}\}$, 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.

arXiv comment14 pages. Correction in Proof of Lemma 4.2 and additional questions in Conclusion