Monochromatic Solutions for Multi-Term Unknowns

Abstract

Given integers $k\ge 2$, $n \ge 2$, $m \ge 2$ and $a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}$, and let $f(z)= \sum _{j=0}^{n}c_jz^j$ be a polynomial of integer coefficients with $c_n>0$ and $(\sum _{i=1}^ma_i)|f(z)$ for some integer z. For a k-coloring of $[N]=\{1,2,\ldots ,N\}$, we say that there is a monochromatic solution of the equation $a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)$ if there exist pairwise distinct $x_1,x_2,\ldots ,x_m\in [N]$ all of the same color such that the equation holds for some $z\in \mathbb {Z}$. Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if $a_i>0$ for $1\le i\le m$, then there exists an integer $N_0=N(k,m,n)$ such that for $N\ge N_0$, each k-coloring of [N] contains a monochromatic solution $x_1,x_2,\ldots ,x_m$ of the equation $a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)$. Moreover, if n is odd and there are $a_i$ and $a_j$ such that $a_ia_j<0$ for some $1 \le i\ne j\le m$, then the assertion holds similarly.