Power Set Width

Probability Level 2

Let X X be a set of subsets of {1,2,3,4} \{1,2,3,4\} such that no element of X X is completely contained in any other element of X X : that is, for any two distinct subsets A,BX A,B \in X, AB A \nsubseteq B and BA B \nsubseteq A .

What is the maximum possible value of X |X| ?

(Bonus: generalize to {1,2,,n} \{1,2,\ldots,n\} .)

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