In 1933, Borsuk stated a conjecture, which has become classical, that the minimum number of parts of smaller diameter into which an arbitrary set of diameter 1 in \(\mathbb{R}^n\) can be partitioned is \(n+1\). In 1993, this conjecture was disproved using sets of points with coordinates 0 and 1. Later, the second author obtained stronger counterexamples based on families of points with coordinates \(-1\), \(0\), and \(1\). We establish new lower bounds for Borsuk numbers in families of this type.

中文翻译：

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特殊类型距离图中 Borsuk 数的边界

1933 年，Borsuk 提出了一个已经成为经典的猜想，即\(\mathbb{R}^n\) 中的任意一组直径 1可以被划分成的较小直径部分的最小数量是\(n+ 1\)。1993 年，这个猜想被坐标为 0 和 1 的点集推翻。 后来，第二作者基于坐标为\(-1\)、\(0\)和\(1\ )的点族得到了更强的反例)。我们为此类家族中的 Borsuk 数建立了新的下限。