Abstract
Let p ∈ ℤ [x] be any polynomial with p(0) =0, k ∈ ℕ and let c1, …, cs ∈ ℤ, s ⩾ k(k + 1), be non-zero integers such that \(\sum {{c_1} = 0} \). We show that for a wide class of coefficients c1, …, cs in every finite coloring \(\mathbb{N} = {A_1} \cup \cdots \cup {A_r}\) there is a monochromatic solution to the equation
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