On monochromatic linear recurrence sequences

Abstract

In this paper we prove some van der Waerden type theorems for linear recurrence sequences. Under the assumption $a_{i-1}\leq a_{i}a_{s-1}$ ($i=2,\ldots,s$), we extend results of G. Nyul and B. Rauf for sequences satisfying $x_i=a_1x_{i-s}+\cdots+a_sx_{i-1}$ ($i\geq s+1$), where $a_{1},\ldots,a_{s}$ are positive integers. Moreover, we solve completely the same problem for sequences satisfying the binary recurrence relation $x_i=ax_{i-1}-bx_{i-2}$ ($i\geq 3$) and $x_1<x_2$, where $a,b$ are positive integers with $a\geq b+1$