給定一個集合，如何構造它的一個儘可能大的子集族，使得子集族中的任意兩個子集之間都不存在包含關係？

01-06

如果要求子集個數盡量多，我想你說的應該是Sperner定理？

A family of sets that does not include two sets X and Y for which X ? Y is called a Sperner family. For example, the family of k-element subsets of an n-element set is a Sperner family. No set in this family can contain any of the others, because a containing set has to be strictly bigger than the set it contains, and in this family all sets have equal size. The value of k that makes this example have as many sets as possible is n/2 if n is even, or the nearest integer to n/2 if n is odd. For this choice, the number of sets in the family is $inom{n}{lfloor frac{n}{2} floor}$ .
Sperner"s theorem states that these examples are the largest possible Sperner families over an n-element set. Formally, the theorem states that, for every Sperner family S whose union has a total of n elements,

$|S| leq inom{n}{lfloor frac{n}{2} floor}$

參考：https://en.wikipedia.org/wiki/Sperner%27s_theorem

最多C(n,n/2)個，在取遍所有n/2元子集時取到。

這是組合中的Sperner定理，在上世紀提出的，已經發展成一套成熟的理論。Sperner定理的實質是子集格中的最大反鏈的勢。