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A Diophantine Ramsey Theorem

Abstract

Let p ∈ ℤ [x] be any polynomial with p(0) =0, k ∈ ℕ and let c1, …, cs ∈ ℤ, sk(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

$${c_1}x_1^k + \cdots + {c_s}x_s^k = \text{p}(y).$$

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Correspondence to Tomasz Schoen.

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Schoen, T. A Diophantine Ramsey Theorem. Combinatorica 41, 581–599 (2021). https://doi.org/10.1007/s00493-020-4482-5

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Mathematics Subject Classification (2010)

  • 11P99
  • 05D10