The famous van der Waerden’s theorem states that if $${\mathbb{N}}$$ N is finitely colored then one color class will contain arithmetic progressions of arbitrary length. The polynomial van der Waerden’s theorem says that if $$p_{1}(x),p_2(x),\ldots ,p_{k}(x)$$ p 1 ( x ) , p 2 ( x ) , … , p k ( x ) are polynomials with integer coefficients and zero constant term and $${\mathbb{N}}$$ N is finitely colored, then there exist $$a,d\in {\mathbb{N}}$$ a , d ∈ N such that $$\big \{a+p_{t}(d):t\in \{1,2,\ldots ,k\} \big \}$$ { a + p t ( d ) : t ∈ { 1 , 2 , … , k } } is monochromatic. There are dynamical, algebraic, and combinatorial proofs of this theorem. In this article we will prove a “near zero” version of the polynomial van der Waerden’s theorem. That is, we will show that for any $$\epsilon >0$$ ϵ > 0 , if $$(0,\epsilon )\cap {\mathbb{Q}}$$ ( 0 , ϵ ) ∩ Q is finitely colored then there exists $$a,d\in {\mathbb{Q}}{\setminus }\{0\}$$ a , d ∈ Q \ { 0 } such that $$\big \{a+p_{t}(d):t\in \{0,1,2,\ldots ,k\}\big \}$$ { a + p t ( d ) : t ∈ { 0 , 1 , 2 , … , k } } is monochromatic.

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Hales-Jewett 定理在零附近的应用

著名的范德瓦尔登定理指出，如果 $${\mathbb{N}}$$ N 是有限颜色的，那么一个颜色类将包含任意长度的等差数列。多项式范德瓦尔登定理说，如果 $$p_{1}(x),p_2(x),\ldots ,p_{k}(x)$$p 1 ( x ) , p 2 ( x ) , ... , pk ( x ) 是具有整数系数和零常数项的多项式并且 $${\mathbb{N}}$$ N 是有限着色的，那么存在 $$a,d\in {\mathbb{N}}$$ a , d ∈ N 使得 $$\big \{a+p_{t}(d):t\in \{1,2,\ldots ,k\} \big \}$$ { a + pt ( d ) : t ∈ { 1 , 2 , … , k } } 是单色的。这个定理有动力学、代数和组合证明。在本文中，我们将证明多项式范德瓦尔登定理的“接近零”版本。也就是说，我们将证明对于任何 $$\epsilon >0$$ ϵ > 0 ，如果 $$(0,