Abstract
Let k,a∈N and let p1,⋯,pk∈Q[n] with zero constant term. We show that for any finite coloring of Q, there are non-zero x,y∈Q such that there exists a color which contains the set{x,xya,x+p1(y),⋯,x+pk(y)} and there are non-zero v,u∈Q such that there exists a color which contains the set{v,v⋅ua,v+p1(u),⋯,v+pk(u)}.
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