The Fibonacci numbers are not 3-accessible

Abstract

A $D$-diffsequence is a sequence of integers $x_1<\dots<x_k$ such that $x_{i+1} -x_i \in D$ for $1 \le i \le k-1$. The set $D$ is called $r$-accessible if every $r$-coloring of the positive integers contains arbitrarily long monochromatic $D$-diffsequences. This note proves that the set of Fibonacci numbers is not 3-accessible, resolving an open problem of Landman and Robertson from 2007.