Gerver-Ramsey共线性问题的改进界

SIAM J. Discret. Math. Pub Date : 2023-03-25 DOI:10.48550/arXiv.2303.14579

T. Lidbetter

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引用次数: 0

摘要

设$S$是$\mathbb{Z}^n$的有限子集。向量序列$(\mathbf{z}_i)$是$S$-walk当且仅当$\mathbf{z}_{i+1} - \mathbf{z}_i$对于所有$i$都是$S$的元素。Gerver和Ramsey在1979年证明了对于$S\子集\mathbb{Z}^3$存在一个无限$S$-walk，其中没有$5^{11}+ 1=48{\small，}828{\small，}126$点共线。在这里，我们使用相同的一般方法，但借助计算机搜索，将绑定提高到$189$。

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Improved Bound for the Gerver-Ramsey Collinearity Problem

Let $S$ be a finite subset of $\mathbb{Z}^n$. A vector sequence $(\mathbf{z}_i)$ is an $S$-walk if and only if $\mathbf{z}_{i+1} - \mathbf{z}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $S\subset \mathbb{Z}^3$ there exists an infinite $S$-walk in which no $5^{11} + 1=48{\small,}828{\small,}126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.

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SIAM J. Discret. Math.

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