论文标题
极端集理论中的链依赖性条件
Chain-dependent Conditions in Extremal Set Theory
论文作者
论文摘要
在极端集理论中，我们通常的目标是找到满足条件的$ n $ emlement集合子集的最大尺寸。条件称为链条依赖性，如果且仅当它与$ n！$全链的交集满足时，它对家庭满意。我们介绍了一种处理此类条件问题的方法，然后展示如何使用它来证明三个经典定理。然后，关于包含两套家庭的家庭的定理证明了\ le | b | $。最后，我们调查了问题，而不是家庭的大小，而是最大化$ \ ell $ chains的数量。我们的方法是在集合（或$ \ ell $ - 链）上定义权重函数，并将其用于涉及完整链的双重计数参数。
In extremal set theory our usual goal is to find the maximal size of a family of subsets of an $n$-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the $n!$ full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems. Then, a theorem about families containing no two sets such that $A\subset B$ and $λ\cdot |A| \le |B|$ is proved. Finally, we investigate problems where instead of the size of the family, the number of $\ell$-chains is maximized. Our method is to define a weight function on the sets (or $\ell$-chains) and use it in a double counting argument involving full chains.
