arXiv 1902.055402019-02-14cs.DMcs.FL

论长词避字格

On long words avoiding Zimin patterns

Arnaud Carayol, Stefan Göller

摘要：

如果一个词的某个中缀是某个非擦除态射下的模式的像，则在该词中遇到模式。如果在每个有限字母表上，每个足够长的单词都会遇到$p$，则模式$p$是不可避免的。Zimin的一个定理以及Bean、Escherichfeucht和McNechte独立的定理指出，$n$不同变量上的模式是不可避免的，当且仅当$p$本身在$n$ -th Zimin模式中遇到。给定一个字母表大小的$k$，我们研究了最小长度$f(n,k)$，使得每个长度为$f(n,k)$的单词遇到$n$ -th Zimin模式。已知$f$的上限是一个指数塔。我们的主要结果表明，即使对于$k=2$，$f(n,k)$也是由$n-3$指数塔下界的。据我们所知，这改进了以前最著名的双指数下限。作为进一步的结果，我们证明了一个双指数上界遇到Zimin模式的阿贝尔意义。

英文摘要：

A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over $n$ distinct variables is unavoidable if, and only if, $p$ itself is encountered in the $n$-th Zimin pattern. Given an alphabet size $k$, we study the minimal length $f(n,k)$ such that every word of length $f(n,k)$ encounters the $n$-th Zimin pattern. It is known that $f$ is upper-bounded by a tower of exponentials. Our main result states that $f(n,k)$ is lower-bounded by a tower of $n-3$ exponentials, even for $k=2$. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense.

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