论文标题
加性和乘法Gower的Ramsey定理
Additive and multiplicative Gower's Ramsey theorem
论文作者
论文摘要
W. T. Gower在$ x_ {k} = \ left \ {\ left（n_ {1}，n_ {2}，\ ldots，n_ {k} \ right）上：使该序列产生的高尔子空间是单色的。对于$ k = 1，$这立即给出有限的总理定理。在本文中，我们将证明，对于$ x_ {k} $的任何有限颜色，存在两个序列$ \ weft \ {\ MathBf {n_ {i}}：i \ in I \ right \} $ and $ \ \ \ left \ left \ left \ left \ apthbf {\ mathbf {\ mathbf {m_ {i} g {i} i \ ims prird prird \ imes i \ right \ imes i \ right i \ right i \ rigr $ \ left \ {\ mathbf {n_ {i}}：i \ in I \ right \} $，以及$ \ left \ left \ {\ mathbf {\ mathbf {m_ {i}}}的所有有限产物的集合：I \ in i \ right \} $。这立即概括了V. Bergelson和N. Hindman的结果，该结果说，对于任何有限的颜色，$ \ Mathbb {n} $，存在两个序列$ \ left（x_ {n} \ right）_ {n} $ $ \ left（x_ {n} \ right）_ {n} $和$ \ left（y_ {n} \ right）_ {n} $的颜色相同。
W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace generated by that sequence is monochromatic. For $k=1,$ this immediately gives the finite sum theorem. In this article we will show that for any finite coloring of $X_{k}$ there exist two sequences $\left\{ \mathbf{n_{i}}:i\in I\right\} $ and $\left\{ \mathbf{m_{i}}:i\in I\right\} $ such that the Gower subspace generated by $\left\{ \mathbf{n_{i}}:i\in I\right\} $ and set of all finite products of $\left\{ \mathbf{m_{i}}:i\in I\right\} $ are in a single color. This immediately generalize a result of V. Bergelson and N. Hindman which says that for any finite coloring of $\mathbb{N}$, there exist two sequences $\left(x_{n}\right)_{n}$ and $\left(y_{n}\right)_{n}$ such that the finite sum and product generated by $\left(x_{n}\right)_{n}$ and $\left(y_{n}\right)_{n}$ are in a same color.
