A note on antichains in the continuous cube

TL;DR

This paper proves that for any dimension, there exist antichains in the continuous cube with maximal Hausdorff measure, confirming a conjecture about their measure bounds.

Contribution

It demonstrates the existence of antichains in the continuous cube that attain the conjectured maximal Hausdorff measure for all dimensions.

Findings

Existence of antichains with maximal Hausdorff measure in [0,1]^n

Confirmation of the conjecture on measure bounds for antichains

Advancement in understanding measure properties of antichains in continuous posets

Abstract

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.