It is well-known that an antichain in the poset $[0,1]^n$ must have measure zero. Engel, Mitsis, Pelekis and Reiher showed that in fact it must have $(n-1)$-dimensional Hausdorff measure at most $n$, and they conjectured that this bound can be attained. In this note we show that, for every $n$, such an antichain does indeed exist.

中文翻译：

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关于连续立方体中反链的注解

众所周知，poset $[0,1]^n$ 中的反链必须具有度量为零。Engel、Mitsis、Pelekis 和 Reiher 证明事实上它必须有 $(n-1)$ 维的 Hausdorff 测度至多 $n$，并且他们推测这个界限是可以达到的。在这篇笔记中，我们表明，对于每个 $n$，这样的反链确实存在。