We introduce the abstract notion of a chain, which is a sequence of $n$ points in the plane, ordered by $x$-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations. We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have $\Omega(9.08^n)$ triangulations. This is a significant improvement over the previous and long-standing lower bound of $\Omega(8.65^n)$ for the maximum number of triangulations of planar point sets.

翻译：我们引入了链的抽象概念,即平面上按x坐标排序的n个点序列,使得任意两个连续点之间的边在三角剖分意义下是不可回避的。我们发展了链结构性质的一般理论,同时对其三角剖分数目建立了普遍认识。我们还描述了一种有趣的新型具体构型,因其与科赫曲线的相似性而称之为科赫链。随后展示基于科赫链的特定构造具有Ω(9.08^n)个三角剖分。这一结果显著改进了之前长期存在的平面点集最大三角剖分数目的下界Ω(8.65^n)。