LYM-不等式在Semiantichains上的推廣

Abstract

設$X$是一個被分割成兩部份$\{ X_1, X_2 \}$的有限集合。若$\cal F$ 為$X$的羃集合□的一個子集合, 且合於下述條件:如果$\cal F$□面任兩 個元素,$A \subset B$則$B -A$不會包含於$X_1$,亦不會包含於$X_2$,那 麼我們就稱 $\cal F$對於分割$\{ X_1, X_2 \}$是一個semiantichain. 我們稱${\cal D}({\cal F}; X_1,X_2)=$ $\sum_{i=0}^n \frac {\mid {\cal F}\cap {X \choose i}\mid}{\mid {X \choose i} \mid}$.為$( X, {\cal F})$的密度和。在這篇論文,我們將研究有關於semiantichain 的密度和的上界及其它一些性質。 Suppose $X$ is a finite set of $n$ elements with the partition $\it X=X_1 \cup X_2$. A family $\cal F \subseteq$ $2^X $ is called a semi-antichain \rm with respect to $\it \{X_1,X_2\}$ if there are no $A,B \in \cal F$ with $A \subset B$ such that $B-A$ is contained in either $\it X_1$ or $\it X_2$. The density-sum of $(X, \cal F)$ is defined to be $\sum_{i=0}^n \frac {\mid {\cal F}\cap {X \choose i} \mid } {\mid {X \choose i} \mid}$. Some properties of the density-sum for semiantichains are studied in this paper.