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- W2967991851 abstract "A emph{chain} in the unit $n$-cube is a set $Csubset [0,1]^n$ such that for every $mathbf{x}=(x_1,ldots,x_n)$ and $mathbf{y}=(y_1,ldots,y_n)$ in $C$ we either have $x_ile y_i$ for all $iin [n]$, or $x_ige y_i$ for all $iin [n]$. We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy [ text{card}(A cap C) le k, , text{ for all chains } , C subset [0,1]^n , , ] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. We show that the $(n-1)$-dimensional Hausdorff measure of a $k$-antichain in $[0,1]^n$ is at most $kn$ and that the bound is asymptotically sharp. Moreover, we conjecture that there exist $k$-antichains in $[0,1]^n$ whose $(n-1)$-dimensional Hausdorff measure equals $kn$ and we verify the validity of this conjecture when $n=2$." @default.
- W2967991851 created "2019-08-22" @default.
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- W2967991851 date "2020-04-01" @default.
- W2967991851 modified "2023-09-24" @default.
- W2967991851 title "On $k$-antichains in the unit $n$-cube" @default.
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- W2967991851 doi "https://doi.org/10.5486/pmd.2020." @default.
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