Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in $\mathcal{A}$. Given reals $\alpha,\beta>0$, we say the weight of $\mathcal F$ is $\alpha\cdot|\mathcal{A}|+\beta\cdot|\mathcal{B}|$. We characterize the minimum weight antichains $\mathcal F$ for any given $n,k,\alpha,\beta$, and we do the same when in addition $\mathcal F$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function.
翻译:以古典的Sperner 和 Kruskal- Katona 的古典理论为基础, 我们调查在 Boolean lattice $B_n美元的所有子集中的反链 $mathcal F$是平的 :\\\ 1, 2,\ dots, n ⁇ $, 这意味着它包含最多连续两个大小的一组, 比如 $mathcal Fámathca{A\ mathcal{B} 美元, 其中 $macal{ A} 美元只包含 $kcal$- subsets, 而 $macal{b> 美元只包含$(k-1) 美元 美元 ; 此外, 我们假设 $\ mathcal{ a} 美元包含第一个 $kmall susubs, 而 $( 1) mac_\\\\ mac) max max 。 max max max max max max macal max max max macal max max max max f cal max max max maxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx