Let p ∈ ℤ [x] be any polynomial with p(0) =0, k ∈ ℕ and let c₁, …, c_s ∈ ℤ, s ⩾ k(k + 1), be non-zero integers such that \(\sum {{c_i} = 0} \). We show that for a wide class of coefficients c₁, …, c_s in every finite coloring ℕ = A₁∪ ⃯ ∪ A_r there is a monochromatic solution to the equation \({c_1}x_1^k + \cdots + {c_s}x_s^k = {\rm{p}}\left(y \right).\).

中文翻译：

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Diophantine Ramsey定理Tomasz Schoen

令p∈ℤ[ X ]是任何多项式其中p（0）= 0，ķ ∈ℕ和让ç ₁，...，c ^_小号∈ℤ，小号⩾ ķ（ķ + 1），是非零的整数，使得\ （\ sum {{c_i} = 0} \）。我们表明，对于宽类系数Ç ₁，...，c ^ _š在每个有限着色ℕ=阿₁ ∪⃯∪甲_{- [R}有一个单色溶液等式\（{C_1} X_1 ^ K + \ cdots + { c_s} x_s ^ k = {\ rm {p}} \ left（y \ right）。\）。