New bounds on diffsequences

Abstract

For a set of positive integers D, a k-term D-diffsequence is a sequence of positive integers a1<a2<…<ak such that ai−ai−1∈D for i=2,3,…,k. For k∈Z+ and D⊂Z+, we define Δ(D,k), if it exists, to be the smallest integer n such that every 2-coloring of {1,2,…,n} contains a monochromatic D-diffsequence of length k. We improve the lower bound on Δ(D,k) where D={2i|i∈Z≥0}, proving a conjecture of Chokshi, Clifton, Landman, and Sawin. We also determine all sets D={d1,d2,…} satisfying ∀i∈Z+,di|di+1 for which Δ(D,k) exists for all k∈Z+. © 2024 Elsevier B.V.