Improved Ramsey-type theorems for Fibonacci numbers and other sequences

Abstract

Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression. We consider two variants of van der Waerden numbers: the numbers $n = n(AP_D,k;r)$, the smallest value where every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression with common difference in $D$, and the numbers $n = \Delta(D,k;r)$, the smallest value $n$ where every $r$-coloring of $\{1,\dots,n\}$ contains a sequence $x_1 < \dots < x_k$ where the differences between consecutive terms are members of $D$. We study the case when $D$ is set of Fibonacci numbers $F$ and give improved bounds for the largest $r$ where $n(AP_F,k;r)$ and $\Delta(F,k;r)$ exist for all $k$. Moreover, we give some computational data on $\Delta(D,k;r)$ for other sets $D$.