论文标题
最终定期单词的拉姆齐（Ramsey）表征
A Ramsey Characterisation of Eventually Periodic Words
论文作者
论文摘要
一个分解$ x = u_1 u_2 \ cdots $ a onphabet $ x $上的无限单词$ x $称为“单色”，对于有限单词的给定着色$ x^*$ on Alphabet $ x $，如果每个$ u_i $都是相同的颜色。 Wojcik和Zamboni证明了$ x $的单词是周期性的，并且仅当每种有限的颜色的$ x^*$都有$ x $的单色分解。另一方面，从拉姆齐定理（Ramsey's Theorem）遵循的是，对于\ textit {any} word $ x $，对于每种有限的颜色，$ x^*$的每种有限颜色都有一个$ x $具有单色分解的后缀。如果每个单词$ u_ {k_1} u_ {k_2} \ cdots u_ {k_n} $，则分解为$ x = u_1 u_2 \ cdots $称为`super-Onochromatic'我们在本文中的目的是证明$ x $的单词最终是定期定期的。我们的主要工具是Ramsey的结果，即可能具有独立兴趣的交替款项。
A factorisation $x = u_1 u_2 \cdots$ of an infinite word $x$ on alphabet $X$ is called `monochromatic', for a given colouring of the finite words $X^*$ on alphabet $X$, if each $u_i$ is the same colour. Wojcik and Zamboni proved that the word $x$ is periodic if and only if for every finite colouring of $X^*$ there is a monochromatic factorisation of $x$. On the other hand, it follows from Ramsey's theorem that, for \textit{any} word $x$, for every finite colouring of $X^*$ there is a suffix of $x$ having a monochromatic factorisation. A factorisation $x = u_1 u_2 \cdots$ is called `super-monochromatic' if each word $u_{k_1} u_{k_2} \cdots u_{k_n}$, where $k_1 < \cdots < k_n$, is the same colour. Our aim in this paper is to show that a word $x$ is eventually periodic if and only if for every finite colouring of $X^*$ there is a suffix of $x$ having a super-monochromatic factorisation. Our main tool is a Ramsey result about alternating sums that may be of independent interest.
