Cross-Sperner系统的改进界
Improved bounds for cross-Sperner systems
摘要：
如果不存在某些$F_i \in \mathcal{F}_i$与某些$F_j \in \mathcal{F}_j$相当的$i \not= j$对，则家族$(\mathcal{F}_{1}, \mathcal{F}_{2} , \cdots , \mathcal{F}_{k}) \in \mathcal{P}([n])^k$的集合是交叉Sperner的。这种族的"大小"的两个自然量度是$\sum_{i = 1}^k |\mathcal{F}_i|$之和$\prod_{i = 1}^k |\mathcal{F}_i|$之积。我们证明了新的上限和下限，这些措施一般$n$和$k \ge 2$，大大提高了以前的最佳界限。特别是，我们构建了一个丰富的家庭的反例猜想的Gerbner，Lemons，帕尔默，Patkós，和Szécsi从2011年。
英文摘要：
A collection of families $(\mathcal{F}_{1}, \mathcal{F}_{2} , \cdots , \mathcal{F}_{k}) \in \mathcal{P}([n])^k$ is cross-Sperner if there is no pair $i \not= j$ for which some $F_i \in \mathcal{F}_i$ is comparable to some $F_j \in \mathcal{F}_j$. Two natural measures of the `size' of such a family are the sum $\sum_{i = 1}^k |\mathcal{F}_i|$ and the product $\prod_{i = 1}^k |\mathcal{F}_i|$. We prove new upper and lower bounds on both of these measures for general $n$ and $k \ge 2$ which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011.
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