Monochromatic Solutions for Multi-Term Unknowns

Abstract

Given integers $$k\ge 2$$kź2, $$n \ge 2$$nź2, $$m \ge 2$$mź2 and $$ a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}$$a1,a2,ź,amźZ\{0}, and let $$f(z)= \sum _{j=0}^{n}c_jz^j$$f(z)=źj=0ncjzj be a polynomial of integer coefficients with $$c_n>0$$cn>0 and $$(\sum _{i=1}^ma_i)|f(z)$$(źi=1mai)|f(z) for some integer z. For a k-coloring of $$[N]=\{1,2,\ldots ,N\}$$[N]={1,2,ź,N}, we say that there is a monochromatic solution of the equation $$a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)$$a1x1+a2x2+ź+amxm=f(z) if there exist pairwise distinct $$x_1,x_2,\ldots ,x_m\in [N]$$x1,x2,ź,xmź[N] all of the same color such that the equation holds for some $$z\in \mathbb {Z}$$zźZ. Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if $$a_i>0$$ai>0 for $$1\le i\le m$$1≤i≤m, then there exists an integer $$N_0=N(k,m,n)$$N0=N(k,m,n) such that for $$N\ge N_0$$NźN0, each k-coloring of [N] contains a monochromatic solution $$x_1,x_2,\ldots ,x_m$$x1,x2,ź,xm of the equation $$a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)$$a1x1+a2x2+ź+amxm=f(z). Moreover, if n is odd and there are $$a_i$$ai and $$a_j$$aj such that $$a_ia_j<0$$aiaj<0 for some $$1 \le i\ne j\le m$$1≤iźj≤m, then the assertion holds similarly.